Essays on the Macroeconomics of Climate Change

Essays on the Macroeconomicsof Climate Change

Johan Gars

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c© Johan Gars, Stockholm, 2012

ISBN 978-91-7447-500-5ISSN 0346-6892

Cover picture: Före stormenc© Fanny Hagdahl Sörebo

Printed in Sweden by PrintCenter US-AB, Stockhom 2012Distributor: Institute for International Economic Studies

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Doctoral DissertationDepartment of EconomicsStockholm University

Abstract

This thesis consists of three self-contained essays dealing with di�erentmacroeconomic aspects of climate change.

Technological Trends and the Intertemporal Incentives for

Fossil-Fuel use analyzes how (the expectations about) the future de-velopments of di�erent kinds of technology a�ect the intertemporal in-centives for fossil-fuel use. Given that fossil-fuel resources are �nite, thedecision of when to extract should be based on the value of fossil-fueluse at di�erent points in time. This means that the expectations aboutthe future state of technology matter for the extraction decisions madetoday. I �nd that improvements in (the expectations about) the futurestate of technologies for alternative-energy generation, energy e�ciencyand total factor productivity (TFP) all increase fossil-fuel use before thechange takes place. The e�ect of changes in the e�ciency of non-energyinputs is the reverse, while the e�ect of changes in fossil-fuel based en-ergy technology is ambiguous. These conclusions are robust to a numberof di�erent possible variations of assumptions. Throughout this chapter,I emphasize the scarcity aspect of the fossil-fuel supply. This seems tobe the crucial assumption. If fossil-fuel supply is, instead, mostly drivenby extraction costs, some results may be reversed.

The Role of the Nature of Damages considers di�erent waysin which climate change can be assumed to a�ect the economy (e.g.,through various damages) and to what extent the choice of how to modelthese climate e�ects matters. In particular, I consider the choice of mod-eling climate impacts as a�ecting productivity, utility or the depreciationof capital.

I carry out my analysis in two di�erent ways. Firstly, under some sim-plifying assumptions, I derive a formula for the optimal tax on fossil-fueluse. The optimal tax at each point in time can be written as a constanttimes current production, where the constant adds up the three di�er-ent types of e�ects. Secondly, I use a two-period model with exogenousclimate to analyze how the allocation of fossil-fuel use over time is af-fected by the e�ects of climate change. I consider two di�erent casesfor the fossil-fuel supply: an oil case, that emphasizes scarcity, and acoal case, that emphasizes extraction costs. I �nd that, for both the oiland coal cases, a decrease in second-period productivity and a worseningof the second-period climate state have the same qualitative e�ects onthe allocation of fossil-fuel use while an increase in the depreciation of

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capital has the opposite e�ect. The e�ects are also very di�erent in thecoal case compared to the oil case. I then ask whether these reactionsto climate change will amplify or dampen climate change. I �nd thatclimate e�ects on productivity or utility will dampen climate change inthe oil case and amplify it in the coal case. The opposite holds for e�ectson capital depreciation.

Indirect E�ects of Climate Change investigates how direct ef-fects of climate change in some countries have indirect e�ects on othercountries going through changing world market prices of goods and �-nancial instruments. When calculating the total e�ects of climate changethese indirect e�ects must also be taken into account.

If climate change decreases the productivity of a country that is anet exporter of a good, the world market price will go up, decreasingthe welfare in countries that are net importers of that good. Financialinstruments can be used to insure against weather related uncertainty.The probability distribution of weather events is expected to change dueto climate change. This means that the world market prices of �nancialinstruments will change as the probability distribution of weather eventschanges. The indirect e�ects going through the price changes of assetswill bene�t or hurt countries depending on whether they are net buyersor net sellers of the assets.

Cost-e�cient mitigation of climate change (reduction of emissions ofgreenhouse gases) requires reductions in all countries. The uneven dis-tribution of the e�ects of climate change poses a problem for agreeingon mitigation e�orts, especially since there seems to be a negative cor-relation between emissions of greenhouse gases and the vulnerability toclimate change. Including the indirect e�ects gives a di�erent distribu-tion of total e�ects which can make it easier or more di�cult to reachagreements depending on whether the indirect e�ects make the coun-tries' interests more or less aligned. The net e�ects will depend on therelation between the direct e�ects and the trade patterns. I argue, basedon a stylized two country example, that trade in goods will tend to makethe countries' interests more aligned while trade in �nancial instrumentswill tend to make the countries' interests less aligned.

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To Ulrika

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Acknowledgments

First, I would like to thank my advisor Per Krusell. For his sharp insightsand relaxed attitude, for always providing excellent guidance while re-specting my take on things; simply, for making the writing of this thesisso much less burdensome and more enjoyable.

I would also like to thank the Department of Economics at Stock-holm University for accepting me into the PhD program and hosting meinitially. I spent most of my time as a PhD student at the Institute forInternational Economic Studies (IIES). I would like to express my sin-cere gratitude to everybody at the institute for making it the inspiringresearch environment that it is. Being part of it has had a profoundimpact on me as a researcher and on the work contained in this thesis.

My joining the IIES coincided with the start-up of the Mistra-SWECIAprogramme on climate change. This gave me the opportunity to dis-cover the �eld of the macroeconomics of climate change alongside someoutstanding researchers: John Hassler, Per Krusell, Conny Olovsson,Torsten Persson and David von Below. It meant that I always had some-one to talk to when I got stuck but also that I could observe how theyapproached a new research �eld. This gave me very valuable insightsinto the research process. Being part of the interdisciplinary Mistra-SWECIA programme also gave me a much broader understanding ofthe climate change issue.

During my time in the PhD program, I have had many great fellowstudents. I would especially like to mention three of them with whom Ihave had many inspiring discussions: David Yanagizawa Drott, DanielSpiro and Gustav Engström.

Throughout my time at the IIES I have received excellent administra-tive and computer support. Christina Lönnblad helped me with editingparts of the thesis. Annika Andreasson helped me with the booking ofthe dissertation and keeping track of all the requirements.

In September 2011 I joined the Beijer Institute. I am very gratefulto all the people there for making me feel very welcome and for makingme look forward to continue working there after the dissertation.

On a more personal note, I would like to mention some of the non-work-related people that are special to me. I would like to thank myparents, Christina and Ulf, for their unconditional support in this andeverything else that I do. My sister, Anna, and her family, for alwaysbeing there. Mikael, thank you for (among other things) building theroom where much of the work in this thesis was carried out. I would liketo thank Emil and Fanny for making everyday life so much more fun.

Finally, Ulrika: thank you for being a constant source of comfort, forhelping me keep things in perspective and for, during the last months ofhard work, giving me glimpses of the life to come.

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Contents

1 Introduction 1

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Technological Trends and the Intertemporal Incentives

for Fossil-Fuel Use 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Model without capital and externalities . . . . . . . . . . 132.3 Two-period model with capital . . . . . . . . . . . . . . . 222.4 Introducing climate change . . . . . . . . . . . . . . . . . 262.5 Model with capital and σY = θ = 1 . . . . . . . . . . . . 552.6 Elastic supply of the alternative-energy input . . . . . . 602.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 672.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . 71References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.A Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 75

3 The Role of the Nature of Damages 95

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 953.2 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . 983.3 Two-period model . . . . . . . . . . . . . . . . . . . . . . 1143.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 133References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353.A Derivatives of a CES production function . . . . . . . . . 1373.B Calculations for the oil case . . . . . . . . . . . . . . . . 137

4 Indirect E�ects of Climate Change 149

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1494.2 Trade in goods . . . . . . . . . . . . . . . . . . . . . . . 1524.3 Insurance against weather variability . . . . . . . . . . . 1674.4 Conclusions and discussion . . . . . . . . . . . . . . . . . 177References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.A Calculations for trade in goods with trading costs . . . . 181

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Chapter 1

Introduction

This thesis consists of three self-contained essays on issues related to themacroeconomics of climate change. The �rst two chapters are relativelysimilar in terms of the questions asked and the models used to answerthem. They both use neoclassical growth models where the world istreated as one large economy. They both consider issues of intertempo-ral incentives for fossil-fuel use and the use of taxation to correct for theexternalities through climate change caused by the burning of fossil fuels.One might say that these two chapters deal with allocation over time.The third chapter, instead, uses models with many countries and consid-ers how e�ects of climate change propagate between countries throughmarket mechanisms. That is, it considers allocation across countries.

Climate change has become a topic of intense public debate in recentyears. One contributing factor to this was the publication of the SternReview (Stern, 2007). The basic mechanisms that are driving climatechange have been known for a long time. More than a hundred yearsago the increase in the global temperature following an increased con-centration of greenhouse gases in the atmosphere was calculated fairlyaccurately. At that time, however, this was not necessarily considered athreat (for instance, the Swedish chemist and physicist Svante Arrheniuswho was one of the pioneers thought, understandably, that a warmerclimate might well be bene�cial). Over time, the problems and risksassociated with climate change have become more and more apparent.

The Intergovernmental Panel on Climate Change was created in1988.1 It has since then published four assessment reports and a �fthis scheduled to be published in 2013 and 2014. The fourth assessmentreport, published in 2007, received much attention and the organization,together with Al Gore, was awarded the 2007 Nobel Peace Prize.

Climate change is not a new topic within economic research either.

1See www.ipcc.ch.

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2 CHAPTER 1. INTRODUCTION

Perhaps the best known economist working with these issues is WilliamNordhaus; he has studied the interaction between the climate and theeconomy since the 1970s. Nordhaus has also developed one of the mostwidely used family of tools that jointly model the economy and theclimate: the DICE/RICE models.2 While his importance for bringingtogether models of the climate and the economy is di�cult to overes-timate, these models (and most other so called IAMs, i.e., integratedassessment models) have a problem: they are highly complex and di�-cult to use for qualitative interpretation. One reason for this is that theyconsist of a large set of equations that can only be solved numerically.

The Mistra SWEdish research programme on Climate, Impacts andAdaptation (Mistra-SWECIA), which I have been a part of, was startedin 2008. One of the main purposes of the macroeconomic modeling partof the programme was to approach the problem somewhat di�erently.The economic part of the models should be based on modern macroe-conomic theory, making the models accessible to mainstream macroe-conomists. The models should also be more transparent.

The work in this thesis very much re�ects this aim for transparency.Rather than using large complex models, the chapters in this thesisexplores qualitative issues using tractable models. I also think that theresults derived in the thesis point to the value of this approach. Whensetting up an integrated assessment model, a number of assumptionsmust be explicitly or implicitly made. These assumptions can completelychange the qualitative behavior of the model.

One important part of any climate-economy model is the supply offossil fuels. Fossil-fuel resources are �nite (at relevant time scales) andthere is a cost of extracting them. An important question is which ofthese aspects of the resource is more important for extraction decisions.If the �niteness, or scarcity, of the resources is more important, com-paring the value of fossil-fuel use at di�erent points in time will be animportant driver behind the intertemporal pattern of extraction. If,instead, the costs of extraction are more important, the extraction de-cisions will be more about weighing current extraction costs against thecurrent value of fossil-fuel use at each point in time.

Another important issue is alternative-energy generation. Large re-ductions in fossil-fuel use without large reductions in material well-beingwill require a rapid increase in the use of energy generated by alternativesources. The way that the alternative-energy generation is modeled canhave signi�cant consequences for the behavior of IAMs. For example, if

2DICE and RICE stands for �Dynamic Integrated model of Climate and the Econ-omy� and �Regional dynamic Integrated model of Climate and the Economy�, respec-tively. See, e.g., Nordhaus & Boyer (2000) for a description of the models.

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the production function is assumed to have some degree of complemen-tarity between energy and other inputs the alternative-energy assump-tion becomes important. If the model abstracts from alternative energy,or if alternative energy is exogenously given, the complementarity be-tween energy and other inputs translates into complementarity betweenfossil fuel and other inputs. If, instead, the capacity for generating al-ternative energy comes from use of inputs such as installed capital forenergy generation, this implies something very di�erent regarding thecomplementarity between fossil fuel and other inputs.

Furthermore, the functional forms for the production and utility func-tions must be speci�ed. A relatively common assumption regarding theproduction function is that energy is combined with other inputs, suchas labor and capital, according to a Cobb-Douglas production function.In both chapters 2 and 3, it can be seen that this assumption, especiallyif combined with the assumption that the utility function is logarithmic,signi�cantly simpli�es the analysis. It can, however, also be seen thatthese assumptions take away some mechanisms that would be present ifmore realistic assumptions were made.

It may not be a very surprising conclusion that the assumptionsmade when building a model a�ects the results that the model delivers.I would, however, argue that the di�erent possibilities that I considerin this thesis lie within the span of model assumptions used and thatconclusions are sometimes drawn that rely on the particular assump-tions made. At the same time, the quantitative basis for making theseassumptions is sometimes weak. Thus, while the analysis in this thesisoften stops at the point where the consequences of making the di�erentpossible assumptions have been determined, this points to fruitful av-enues for future research. Quantitative analysis of these possible choicesis needed to �nd out what the �right� assumptions are and the quantita-tive consequences for model output such as optimal taxes on fossil-fueluse must be determined.

Chapter 2, Technological Trends and the Intertemporal In-

centives for Fossil-Fuel Use, analyzes how (the expectations about)the future developments of di�erent kinds of technology a�ect the in-tertemporal incentives for fossil-fuel use. Given that fossil-fuel resourcesare �nite, the decision of when to extract should be based on the valueof fossil-fuel use at di�erent points in time. This means that the ex-pectations about the future value of fossil-fuel use matters also for theextraction decisions made today. The future development of technologyis an important determinant of this future value. The literature on theGreen Paradox (see van der Werf and di Maria, 2011, for a survey of thisliterature) has recognized the importance of this aspect of the fossil-fuel


supply for the e�ects of policies aimed at reducing the emissions of CO2

from the burning of fossil fuels. What is found in this literature is thatannounced policies that reduce the future value of fossil-fuel use will tendto increase the current amount of fossil-fuel use and thereby potentiallyexacerbate the problem of climate change. The commonly discussedpolicies are announcements of higher future taxes on fossil-fuel use orinvestments that will increase the future supply of alternative energy.

The topic of this chapter is to consider how (the expectations about)the future developments of a wider range of technologies a�ect the in-tertemporal incentives for fossil-fuel use. The technology trends thatI consider are technology for: alternative-energy generation, fossil-fuelbased energy generation, energy savings, productivity of other (com-plementary) inputs, i.e., labor and sometimes capital, and total factorproductivity (TFP). The analysis in this chapter is carried out using neo-classical models. I use these models to determine the e�ect of a futurechange in the state of each of the technologies on the path of fossil-fueluse. The general conclusion is that improvements in (the expectationsabout) the future state of technologies for alternative-energy generation,energy e�ciency and TFP all increase fossil-fuel use before the changetakes place. The e�ect of changes in the e�ciency of non-energy in-puts is the reverse, while the e�ect of changes in fossil-fuel based energytechnology is ambiguous. These conclusions are robust to a number ofdi�erent possible assumptions. Thus, the e�ects of changes in the fu-ture technology for alternative-energy generation and energy e�ciencycon�rm the �ndings in the Green Paradox literature.

The analysis indicates that the joint e�ects of all technology trendsshould be considered rather than looking at one type of technology inisolation. In reality technology trends are the results of research. In-creasing spending on one type of research will typically have e�ects alsoon the amount of research on other types technologies, e.g., throughcrowding out.

Throughout this chapter, I emphasize the scarcity aspect of the fossil-fuel supply. This seems to be the crucial assumption. If fossil-fuel supplyis, instead, mostly driven by extraction costs, some results may be re-versed.

Chapter 3, The Role of the Nature of Damages, considersdi�erent ways in which climate change can be assumed to a�ect theeconomy (e.g. through various damages) and to what extent the choiceof how to model these climate e�ects matters. The most common wayto introduce the e�ects of climate change into economic models is to as-sume that it a�ects productivity or utility. Some of the expected e�ectsof climate change, e.g., storms and �oods, rather destroy the capital

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stock. Modeling the e�ects of a climate change as increased deprecia-tion of capital therefore seems plausible. In this chapter I consider towhat extent it matters whether climate is assumed to a�ect productivity,utility or the depreciation of capital.

I carry out my analysis in two di�erent ways. Firstly, under somesimplifying assumptions, I derive a formula for the optimal tax on fossil-fuel use. The optimal tax at each point in time can be written as aconstant times current production, where the constant adds up the threedi�erent e�ects that climate change has on the economy. Golosov et al.(2011) derive a similar formula for the optimal tax when consideringonly climate change e�ects on productivity. The formula derived inchapter 3 can therefore be seen as a generalization of that formula. Theassumptions I make in order to derive the formula are also similar.

Secondly, I use a two-period model with exogenous climate to analyzehow the allocation of fossil-fuel use over time is a�ected by the e�ects ofclimate change. I consider two di�erent cases for the fossil-fuel supply:an oil case, where the resources are �nite but I abstract from extractioncosts, and a coal case, where I abstract from the �niteness of the resourcebut extraction requires the use of inputs. I �nd that, for both the oiland coal cases, a decrease in second-period productivity and a worseningof the second-period climate state have the same qualitative e�ects onthe allocation of fossil-fuel use while an increase in the depreciation ofcapital has the opposite e�ect. The e�ects are also very di�erent in thecoal case compared to the oil case.

In the second part of this chapter, I treat climate as exogenous. Thederived e�ects are still indicative of how a decentralized equilibriumwould respond to expected climate change. I then ask whether thesereactions to climate change will amplify or dampen climate change. An-swering this question requires assumptions about how to best representthe e�ects of climate change in a two-period model and what constitutesampli�cation of climate change in the oil and coal cases, respectively.Under the interpretation I choose, climate e�ects on productivity orutility will dampen climate change in the oil case and amplify it in thecoal case. Conversely, climate e�ects on the depreciation of capital willamplify climate change in the oil case, at least if the supply of alternativeenergy is exogenously given, but dampen it in the coal case.

Chapter 4, Indirect E�ects of Climate Change, investigateshow direct e�ects of climate change in some countries have indirect ef-fects on other countries going through changing world market prices ofgoods and �nancial instruments. The direct e�ects of climate changeare expected to di�er a great deal across di�erent countries. However,since the economies of countries are interconnected in various ways the


direct e�ects will be propagated between countries through market mech-anisms. This means that when calculating the total e�ects of climatechange these indirect e�ects must also be taken into account.

In this chapter I consider two such channels: trade in goods andtrade in �nancial instruments. For both of these channels the indirecte�ects go through changing world-market prices of goods and �nancialinstruments. If climate change decreases the productivity of a countrythat is a net exporter of a good, the world market price will go up,decreasing the welfare in countries that are net importers of that good.Weather events cause uncertainty. Financial instruments can be usedto decrease this uncertainty by o�ering insurance against bad outcomes.The probability distribution of weather events is expected to change dueto climate change. This means that the world market prices of �nancialinstruments will change as the probability distribution of weather eventschanges. The indirect e�ects going through the price changes of assetswill bene�t or hurt countries depending on whether they are net buyersor net sellers of the assets.

Climate change depends primarily on total global emissions of green-house gases while the geographical source of the emissions are largelyirrelevant. This means that cost-e�cient mitigation of climate change(reduction of emissions of greenhouse gases) requires reductions in allcountries. The uneven distribution of the e�ects of climate change posesa problem for e�cient mitigation since countries willingness to partici-pate in mitigation e�orts can be expected to be closely related to thecosts from climate change they are expected to su�er. This is made worseby the fact that there seems to be a negative correlation between emis-sions of greenhouse gases and the vulnerability to climate change. Sincethe indirect e�ects of climate change will give a di�erent distributionof the total e�ects compared to the distribution of direct e�ects, theseindirect e�ects can make it easier or more di�cult to reach agreementsabout mitigation e�orts depending on whether the indirect e�ects makethe countries' interests more or less aligned. The net e�ects will dependon the relation between the direct e�ects and the trade patterns. I argue,based on a stylized two country example, that trade in goods will tendto make the countries' interests more aligned while trade in �nancialinstruments will tend to make the countries' interests less aligned.

References

Golosov, M., J. Hassler, P. Krusell & A. Tsyvinski, 2011, "Optimal Taxeson Fossil Fuel in General Equilibrium", NBER Working Paper 17348,http://www.nber.org/papers/w17348.

Nordhaus, W. & J. Boyer, 2000, Warming the World: Economic Modelsof Global Warming, MIT Press, Cambridge, MA.

Stern, N., 2007, The Economics of Climate Change: The Stern Review,Cambridge University Press, Cambridge, UK.

Van der Werf, E. & C. Di Maria, 2011, �Unintended Detrimental E�ectsof Environmental Policy: The Green Paradox and Beyond.� CESifoWorking Paper Series No. 3466.Available at SSRN: http://ssrn.com/abstract=1855899

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Chapter 2

Technological Trends and the

Intertemporal Incentives for

Fossil-Fuel Use

2.1 Introduction

In order to avoid risks of serious negative e�ects of climate change, largereductions of emissions of greenhouse gases are discussed. In order toaccomplish this without large reductions in economic growth, the use offossil fuel must rapidly be replaced by energy from alternative sources.However, the �niteness of the fossil-fuel resources introduces a kind ofintertemporal incentive for extraction that can give unexpected side ef-fects from investments in alternative energy technology. This is knownas the green paradox. In this chapter I will extend previous research toconsider more generally how technological trends a�ect the intertempo-ral incentives for fossil-fuel extraction.

The Copenhagen accord states that the countries should aim to ful�llthe two degree target, meaning that the global mean temperature shouldnot be allowed to increase more than two degrees above pre-industriallevels. This requires very large reductions of the emissions of greenhousegases, of which CO2, from the burning of fossil fuels, is one of the mostimportant ones. As an example, the European commission's roadmap formoving to a competitive low carbon economy in 2050 (European Com-mission 2011) says that the developed countries should reduce emissionsof greenhouse gases by 80-95% by 2050 in order to reach the two degreetarget. At the same time it seems that, at least in the short run, there islow substitutability between energy and other inputs (see, e.g., Hassleret al. 2011). This implies that, in order to come anywhere near the twodegree target, without seriously hurting economic activity, technological

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10 CHAPTER 2. TECHNOLOGY AND FOSSIL-FUEL USE

change is needed.Fossil fuels are, on relevant time scales, non-renewable and extracted

from a �nite supply. The consequences of this, for the extraction de-cisions of fossil-fuel extracting �rms, was �rst analyzed by Hotelling(1931). Forward looking and pro�t maximizing fossil-fuel resource own-ers should extract in such a way that the marginal (discounted) pro�tsfrom extraction is the same in all periods where extraction is positive.This means that any changes in the future pro�tability of fossil-fuelextraction should a�ect the extraction decisions already today. This im-plies that any decrease in the future pro�tability of fossil-fuel extractionshould lead to increased fossil-fuel extraction in the short run and thatannouncements about policies aimed at reducing emissions of greenhousegases in the future can lead to increased emissions in the short run. Sinn(2008) considered such e�ects and coined the term Green Paradox. Forthe case of an increasing tax on fossil-fuel use, Sinclair (1992) found thatthis increases emissions in the short run. In the terminology of Gerlagh(2011), the Weak Green Paradox refers to a situation where changes inexpectations about future taxation, or improvements in the future stateof alternative-energy technology, generated by an ambition to reduceemissions of green house gases, counter-productively increase emissionsin the short run. In this chapter I will mainly consider e�ects similarto the Weak Green Paradox. The Strong Green Paradox refers to a sit-uation where, over the long run, the e�ects of climate change becomeworse as a consequence of regulation aimed at reducing climate change.A number of papers, for example Gerlagh (2011) and van der Ploeg andWithagen (2012), have further investigated these mechanisms. Van derWerf and Di Maria (2011) provide an overview of this recent literature.

Thus the literature on the green paradox is concerned with howchanges in expectations about future development of alternative-energytechnology, or future taxation, a�ect fossil-fuel use. The result most rel-evant for this chapter is that an improvement in the future availabilityof alternative-energy increases fossil-fuel use in the short run. Motivatedby these studies, the present chapter addresses a broad question: whatare the e�ects of the path of technological development on fossil-fuel use?I also extend the analysis to include many di�erent kinds of technology.The technology trends that I will study are technology for alternative-energy generation, fossil-fuel based energy generation, energy savings,productivity of other (complementary) inputs, i.e., labor and sometimescapital, and general TFP.

I carry out the analysis in the framework of a neoclassical modelwith fossil fuel as a non renewable resource. In some parts of the chapter,fossil-fuel use causes climate change that a�ects future productivity. The

2.1. INTRODUCTION 11

model can be seen as a variation of the model used by Dasgupta andHeal (1974). Compared to that model, I add climate change caused bythe use of fossil fuels as well as the use of alternative-energy sources.

By specifying a general, nested, CES production function, and a CESutility function, I can see how the e�ects of changes in di�erent tech-nology trends depend on the parameters of the production and utilityfunctions. There are two parameters in the production function. One pa-rameter determines the degree of complementarity between energy andother inputs (where the other inputs are labor and sometimes also capi-tal) and the other parameter determines the degree of complementarity(or, rather, substitutability) between di�erent energy sources.

Starting with a model without capital or any climate change relatedexternalities, a set of reasonable assumptions about the values of theparameters allows me to unambiguously determine the e�ect of changesin all the considered technology trends except for the technology forfossil-fuel based energy generation. For the other technology factors,an increase in the future state of TFP, energy-saving technology andalternative-energy technology increases fossil-fuel use in the short run.An increase in the future state of the productivity of the complementaryinputs decreases fossil-fuel use in the short run. The assumptions thatallow me to derive these results are that the CES utility function hasat least logarithmic curvature, that there is a signi�cant degree of com-plementarity between energy and other inputs and that di�erent energysources are close substitutes. This analysis is carried out in section 2.2.The rest of the chapter then considers various extensions and investigatesthe robustness of the basic results to these extensions. It turns out thatthe qualitative results are quite robust to these other aspects. The onlycase in which the basic results do not hold is when alternative-energygeneration uses an endogenously determined input that is extracted us-ing labor. The e�ect of a change in the labor intensive technology is thenambiguous. In summary, considering the e�ects of changes in the futurestate of the technology for alternative-energy generation and energy-saving technology, I obtain a rather general Weak Green Paradox.

Throughout, I treat the technology trends as exogenous. In reality,technological development is driven by research activity. To the extentthat research uses some scarce resources, increased research on one typeof technology will tend to crowd out other types of research. The resultsderived in this chapter implies that it matters what kind of research iscrowded out. The mechanism of the green paradox will be reinforced orweakened by this crowding out depending on what type of other researchis crowded out.

When discussing how fossil-fuel resource owners react to changes in


expectations about future technology this implies an interpretation interms of a market outcome. However, if this reaction is also the optimalreaction, this should not be much of a problem. I therefore, when theydo not coincide, solve for both the market outcome and the planningsolution. I �nd that the change in the planner solution goes in thesame direction as in the decentralized outcome. This implies that it isinteresting to look more at how optimal taxation and the welfare gainsfrom taxation change if the future state of technology changes. I �ndthat if the change in the future state of technology is such that fossil-fueluse increases in the short run, then in the short run the tax rate that canbe used to implement the optimal solution in a competitive equilibriumwith taxation increases too. Regarding the welfare e�ects of taxation, Idemonstrate that these could go either way.

In parts of the chapter, I abstract from capital accumulation to sim-plify the analysis. I check the robustness to including capital in twodi�erent ways. In section 2.3, I consider a two-period model with capi-tal and relatively general forms of the utility and production functions.I also show, in section 2.5, that if capital combines with other inputsas in a Cobb-Douglas production function, if utility is logarithmic andif capital depreciates fully between periods, then almost all the derivedresults will apply equally well to a model with capital. Throughout thechapter, I will assume that fossil fuel is costlessly extracted from a giventotal supply. This is a strong assumption and I will discuss it further insection 2.7.

The rest of the chapter is organized as follows. I start in section 2.2by setting up a model without capital and without externalities. Ini-tially, production in each period just depends on fossil-fuel use and a setof exogenously given variables. The exogenous variables are then speci-�ed as other inputs and technology factors. This formulation allows meto investigate how the results depend on particular parameters in theproduction and utility functions. After that, in section 2.3, I look at atwo-period model with capital but without externalities. Then, in sec-tion 2.4, I introduce externalities in the form of climate change. Withexternalities in the model, I need to distinguish between a decentralizedequilibrium, section 2.4.2, and the planner solution, section 2.4.3. Withexternalities, I can also discuss optimal taxation, in section 2.4.4, andthe welfare gains from taxation, in section 2.4.5. After that, in section2.5, I show that for the special case where energy and the other inputsare combined into �nal goods according to a Cobb-Douglas productionfunction, utility is logarithmic and capital depreciates fully, the solu-tions simplify a lot and almost all the result derived for models withoutcapital hold also with capital. In all previous sections, the supply of al-

2.2. MODEL WITHOUT CAPITAL AND EXTERNALITIES 13

ternative energy has been assumed to depend on the level of technologyfor alternative-energy generation and an exogenously given amount ofan alternative-energy input, implying that improved technology imme-diately transforms into increased use of alternative energy. In section2.6 I, instead, assume that the alternative-energy input is endogenouslydetermined based on the resources required to provide it. Finally, thechapter is concluded with a discussion of the derived results and theassumptions made.

2.2 Model without capital and externalities

This section contains much of the basic intuition underlying the resultsof this chapter. I will start by setting up a model where the only en-dogenously determined variables are the amounts of fossil-fuel use ineach period. Initially, production will depend on fossil-fuel use and aset of abstract, exogenously given, variables. I will consider how vary-ing the exogenously given variables a�ects the equilibrium allocation offossil-fuel use. This demonstrates the basic mechanisms involved thata�ect the incentives for intertemporal allocation of fossil-fuel use. Af-ter that, I specify a speci�c production and utility function so that Ican determine the e�ects of changing particular technology factors andexogenously given inputs.

2.2.1 Model setup

Fossil fuel is costlessly extracted from a �xed supply. Let the amount offuel burned in period t be Bt and the amount of fuel left in the groundat the beginning of period t be Qt.

The constraint on the total available amount of fossil fuel can thenbe written

∞∑t=0

Bt ≤ Q0. (2.1)

Production in a period depends on the amount of fossil fuel used andon a set of exogenously given variables Γ

Y = F (B; Γ).

The production function is assumed to have the properties

∂F

∂B> 0 and

∂2F

∂B2< 0. (2.2)

If the production function also ful�lls the condition

limB→0+

∂F

∂B=∞, (2.3)


then fossil-fuel use will be strictly positive in each period. I will alsoassume that the variables are de�ned so that production depends posi-tively on each variable in Γ.

Consumption is equal to production in each period Ct = Yt. Prefer-ences are given by

∞∑t=0

βtU(Ct),

where β is the discount factor and the period utility function is assumedto have the properties

U ′(C) > 0 and U ′′(C) < 0. (2.4)

2.2.2 Equilibrium

Without externalities in the form of climate change, the planner solutionwill coincide with the competitive equilibrium and I will therefore solvethe planner problem.

The planner problem is to maximize utility given the constraint onthe total amount of available fossil fuel:

max{Bt}∞t=0

∞∑t=0

βtU(Yt) s.t. Bt ≥ 0∀t and∞∑t=0

Bt ≤ Q0.

The Lagrangian of this problem is

L =∞∑t=0

βtU(Yt) + λ

[Q0 −

∞∑t=0

Bt

]+∞∑t=0

µtBt.

The �rst order condition with respect to Bt is

βtU ′(Ct)∂Yt∂Bt

= λ− µt,

where λ > 0 if the constraint on total available fossil fuel binds andµt > 0 if Bt ≥ 0 binds.

Without externalities, and given assumption (2.2), the constraint onthe total supply of fossil fuel will always bind and λ > 0. Assuming thatBT > 0, for some T , the equilibrium condition can be written

βtU ′(Ct)∂Yt∂Bt

≤ U ′(CT )∂YT∂BT

and∞∑t=0

Bt = Q0

with equality whenever Bt > 0. If the production function ful�lls as-sumption (2.3), Bt > 0 for all t.


In essence, what the solution does is that it equalizes the marginalvalue of fossil-fuel use over time.

De�neH(B; Γ) = U ′(Y )

∂Y

∂B. (2.5)

That is, H is the marginal value, in terms of utility, of fossil-fuel use.With this de�nition, assuming that BT > 0, the equilibrium is charac-terized by the two conditions:

βtH(Bt; Γt) = βTH(BT ; ΓT ) for all t such that Bt > 0 (2.6)

and∞∑t=0

Bt = Q0. (2.7)

2.2.3 Changes in the exogenous variables

I will now show how changes in the exogenously given variables a�ectthe equilibrium allocation of fossil-fuel use. If ΓT changes, for T suchthat BT > 0, this will change the marginal value of fossil-fuel use in thatperiod. Since the equilibrium allocation requires that the marginal valueof fossil-fuel use be the same in all periods, and since fossil-fuel use isthe only endogenously determined variable, fossil-fuel use will change inreaction to the change in ΓT in order to equalize the marginal value.

The e�ect on the marginal value of fossil-fuel use of changes in fossil-fuel use is

∂H

∂B= U ′′(Y )

(∂Y

∂B

)2

+ U ′(Y )∂2Y

∂B2.

Under assumptions (2.2) and (2.4)

∂H

∂B< 0. (2.8)

Thus everything else equal, an increase in the fossil-fuel use in a perioddecreases the marginal value of using fossil fuel in that period.

Changes in ΓT change the relative values of using fossil fuel in periodT compared to the value of fossil-fuel use in other periods. If the changein ΓT increases the value of fossil-fuel use in period T , this should lead toa redistribution of fossil-fuel use toward period T . Similarly, a change inΓT that decreases the value of fossil-fuel use in period T should lead toa redistribution of fossil-fuel use away from period T . This is formalizedin the following proposition.

Proposition 2.1. Assume that the sequence {Γt}∞t=0 induces the se-quence {Bt}∞t=0 of fossil-fuel use. Consider two periods t and T such


that T 6= t, Bt > 0 and BT > 0 and a change in XT , de�ned as one ofthe variables in ΓT . Then

Sgn

(dBT

dXT

)= Sgn

(∂HT

∂XT

)and Sgn

(dBt

dXT

)= Sgn

(−∂HT

∂XT

).

Proof. In the induced outcome, (2.6) holds in all periods where there isstrictly positive fossil-fuel use. If the variable XT is varied, the conditionis no longer ful�lled in that period if the fossil-fuel use is unchanged.

If ∂HT∂XT

> 0, (2.8) implies that BT has to increase to still satisfythe equilibrium condition. This would mean that the constraint on thetotal supply of fossil fuel is violated. Using (2.8) again, changing thefossil-fuel use in one period, while keeping all variables constant andwhile maintaining the equilibrium condition (2.6) means changing thefossil-fuel use in all periods in the same direction. This means that thefossil-fuel use in all periods should be decreased until the constraint ontotal supply of fuels is ful�lled. So the net e�ect will be that fossil-fueluse is decreased in all periods with Bt > 0 except in period T where thenet e�ect will be to increase the fossil-fuel use.

The case ∂HT∂XT

< 0 is the mirror image of the previous case.

The following corollary follows directly from this proposition:

Corollary 2.1. Consider two sequences of parameters {ΓIt}∞t=0 and {ΓIIt }∞t=0

with corresponding induced fossil-fuel use {BIt }∞t=0 and {BII

t }∞t=0 respec-tively. Assuming that ΓIt = ΓIIt for all t < T and that for t ≥ T ,H(BI

t ; ΓIIt ) ≤ H(BIt ; ΓIt ), then B

It ≥ BII

t for all t < T (and vice versa).

Proof. Follows from proposition 2.1

Thus, if the expectations in period 0 about the future change in sucha way that, from period T and onwards, the value of using fossil fuel willdecrease, there will be increased fossil-fuel use in all periods before T . Ifthe change in expectations is such that the future value of using fossil fuelwill increase, fossil-fuel use will decrease in all periods before T . Here thechange in the value of fossil-fuel use comes from changes in the exogenousvariables in the production function. In a decentralized equilibrium withtaxation, credible announcements about the future taxes on fossil-fueluse will have a very similar e�ect since it a�ects the relative pro�tabilityof extracting fossil fuel in di�erent time periods.

Note that corollary 2.1 is not su�cient for concluding how Bt willchange in individual periods for t ≥ T . This is because the changes indriving variables in that period in the other periods will tend to movefossil-fuel use in opposite directions.


The proposition and corollary describe the e�ects of changes in XT

(which is one of the variables in ΓT ) through how this change a�ects themarginal value of fossil-fuel use, HT , in period T . The e�ect of changesin XT on HT can be divided into two separate e�ects according to thefollowing derivative:

∂H

∂X=

∂

∂X

[U ′(Y )

∂Y

∂B

]= U ′′(Y )

∂Y

∂X

∂Y

∂B+ U ′(Y )

∂2Y

∂X∂B. (2.9)

The �rst term captures that the change in X a�ects production directlyand thereby a�ects the marginal value of consumption. Since, by as-sumption, U ′′(Y ) < 0, ∂Y

∂X> 0 and ∂Y

∂B> 0, this e�ect is always negative,

capturing that increased consumption decreases the marginal value ofconsumption. The second term captures that the change in X also canhave an e�ect on the marginal productivity of fossil fuel.

From this discussion it follows that if an increase in X decreasesthe marginal product of fossil fuel, then it unambiguously decreases themarginal value of using fossil fuel in that period. If, instead, an increasein X increases the marginal product of fossil fuel, the total e�ect isambiguous and the sign depends on the relative strength of the e�ects.

In sum, the conclusion is that changes in XT that increase the valueof fossil-fuel use in period T will lead to an increase in fossil-fuel use inthat period and a decrease in fossil-fuel use in all other periods. Changesin XT that decreases the value of fossil-fuel use will have the oppositee�ect. The e�ects of a change in Xt on the value of fossil-fuel use inperiod T is given by (2.9).

2.2.4 Interpretation of Γ

In order to say something more concrete about the e�ects of changing thefuture state of di�erent technologies, I will now be speci�c about whatthe variables in Γ are and what the production and utility functions looklike.

There are three inputs to production. These are labor, L, which isassumed to be exogenously given, fossil fuel and an alternative-energyinput, S, which also is exogenously given.

In addition to this, there are �ve technology factors:

1. AY is general TFP

2. AL is labor-intensive technology

3. AE is general energy-saving technology

4. AB is fossil-fuel based technology


5. AS is alternative-energy technology

The technology factors are also assumed to develop exogenously.Under these assumptions, the only variable that is endogenously de-

termined is fossil-fuel use.De�ne the vector of variables

Γ = (L, S,AY , AL, AE, AB, AS).

The e�ects of changing the di�erent variables in Γ will depend on theshape of the utility and production functions. Assume the utility func-tion

U(C) =C1−θ − 1

1− θ, (2.10)

a functional form that is needed for generating outcomes with balancedgrowth, and the production function

F (B; Γ) = AY

[γL (ALL)

σY −1

σY + γE (AEYE)σY −1

σY

] σYσY −1

, (2.11)

where

YE =[γB (ABB)

σE−1

σE + γS (ASS)σE−1

σE

] σEσE−1

(2.12)

is a composite energy good that is produced from fossil fuel and thealternative-energy source.

Note that there is one more technology factor here than is strictlynecessary. Any combination of technology factors could be achievedwith a smaller set of technology factors. For example, any combinationof AB, AE and AS can be achieved by setting any one of them equal to1 and then adjusting the other two accordingly. There are at least tworeasons for maintaining this redundancy. Firstly, each of the technologyfactors translates into measures that are commonly referred to: there arefrequent discussions about energy e�ciency, AE as well as technologiesfor di�erent kinds of energy generation AB and AS. Secondly, looking atthe di�erent technology factors, the e�ect of an arbitrary combinationof changes in AB and AS will sometimes be ambiguous while the signof the e�ect of the particular combination of changes in AB and ASthat corresponds to a change in AE is unambiguously determined by theassumptions made below.

The e�ects will also depend on the parameters in the utility andproduction functions. In (2.11), σY gives the substitutability betweenenergy and other inputs (here labor) while in (2.12), σE gives the sub-stitutability between the di�erent energy sources. It seems reasonable


that energy and other inputs are not very close substitutes while dif-ferent energy sources are close substitutes. This leads to the followingassumptions about parameter values:

σY ≤ 1 < σE. (2.13)

Note that for a �nite σE, this production function ful�lls assumption(2.3) and fossil-fuel use will be strictly positive in all periods. In thelimit as σE →∞, the di�erent energy sources become perfect substitutesand fossil-fuel use will typically only be positive in a �nite number ofperiods.

For the utility function, Layard et al. (2008) �nd that θ lies between1.2 and 1.3. I will assume here that

θ ≥ 1. (2.14)

Furthermore the following assumption will be made:

1

σY≥ θ. (2.15)

This assumption says that the complementarity between energy andother inputs is strong (at least in relation to the curvature of the utilityfunction). Hassler et al. (2011) estimate the elasticity between energyand a Cobb-Douglas composite of capital and labor to be about 0.005.One can expect that the elasticity is larger the larger the time periodbut assuming (2.15) still seems reasonable.

To simplify the notation a bit, let

GL = γL (ALL)σY −1

σY , GE = γE (AEYE)σY −1

σY ,

GB = γB (ABB)σE−1

σE , GS = γS (ASS)σE−1

σE .(2.16)

Using the functional form of the utility function (2.10), the derivativeof H with respect to parameter X (again de�ned as one of the variablesin Γ) is

∂H

∂X= U ′′(C)FXFB + U ′(C)FXB = U ′(Y )

[FXB − θ

FXFBY

], (2.17)

where the subscripts to F refers to partial derivatives.Since U ′(C) > 0, this expression will have the same sign as the last

parenthesis. For qualitative results regarding increases or decreases infossil-fuel use it is the sign that is important. Using (2.11) and (2.12),the expression (2.17) can be calculated for the di�erent variables X. In


wrt sign of (2.17)AY 1− θAL

1σY− θ

AEσY −1σY

GL + (1− θ)GE

AB

(σY −1σY

GL + (1− θ)GE

)GB + σE−1

σEGS

AS, S(

1σE− 1

σY

)GL +

(1σE− θ)GE

Table 2.1: Derivative signs

table 2.1 expressions with the same sign as (2.17) are collected. Thecalculations can be found in appendix 2.A.1.

The results under assumptions (2.13), (2.14) and (2.15) can be sum-marized in the following proposition.

Proposition 2.2. Assume that (2.13), (2.14) and (2.15) hold and thatBt > 0 and BT > 0 for t 6= T . Then

dBt

dAL,T≤ 0 and

dBT

dAL,T≥ 0

anddBt

dXT

≥ 0 anddBT

dXT

≤ 0 for X ∈ {AY , AE, AS, S},

while the signs of the e�ects of changes in AB,T are ambiguous.

Proof. Follows from proposition 2.1 and table 2.1.

These results can be intuitively understood in terms of the two e�ects(described in (2.9)) of changing the marginal product of fossil fuel andchanging the marginal utility of consumption. The way that the exoge-nous variables are de�ned, increasing them always increases production,and therefore also consumption. This decreases the marginal utility ofconsumption which has a negative e�ect on the marginal value of usingfossil fuel. The strength of this e�ect is determined by θ.

Increasing AY increases the marginal product of fossil fuel. The rel-ative strength of the decrease in marginal utility and the increase in themarginal product of fossil fuel is determined by θ and under assumption(2.14) the net e�ect is negative.

Increasing AL increases the marginal product of fossil fuel. Thestrength of this e�ect is determined by the degree of complementar-ity as measured by σY . So the sign of the net e�ect depends on therelative strength of the e�ects. Under assumption (2.15) the net e�ectis positive.


Increasing AE generates a direct positive e�ect on the marginal pro-ductivity of fossil fuel but also a negative e�ect on the marginal value ofenergy, since the amount of energy increases relative to to the amountof complementary inputs, and consumption. Under assumptions (2.13)and (2.14) the negative e�ect will always dominate.

When increasing the supply of alternative energy ASS, there will bea negative e�ect on the marginal value of consumption and the marginalproductivity of energy (since it increases the amount of energy comparedto the amount of complementary inputs). It does, however, also have apositive e�ect on the marginal product of fossil fuel in the productionof the composite energy good. For the parameter assumptions here, thenegative e�ects dominate.

The e�ect of improving the state of the fossil-fuel based technol-ogy AB is not unambiguously determined by the parameter assumptionsmade. Increasing AB has a direct e�ect of increasing the marginal prod-uct of fossil fuel. It also decreases the marginal product of the com-posite energy good and the marginal value of consumption. Which ofthese e�ects will dominate depends on the values of the variables. Un-der assumptions (2.13) and (2.14) the �rst term in the derivative will benegative while the second term will be positive. Some further insight canbe gained by �xing GE and GL, implying that Y is also �xed. The rel-ative importance of fossil-fuel based versus alternative energy can thenbe varied (that is vary GB and GS in such a way that GE remains �xed).From the expression it can be seen that if a large share of energy comesfrom fossil fuel (GB >> GS) the net e�ect will be negative. In this case,varying AB is similar to varying AE. If, on the other hand, a large shareof the energy comes from alternative-energy sources (GS >> GB), thenthe net e�ect will be positive. Assuming that alternative energy will be-come more important relative to fossil-fuel based energy over time, thee�ect of an increase in the future value of AB on the value of fossil-fueluse will tend to be negative (giving an increase in the short run fossil-fueluse) if the change occurs soon, while it will tend to be positive (givinga decrease in the short run fossil-fuel use) if the change occurs in thedistant future.

In conclusion, the sign of the e�ect of changing the future state oftechnology depends on which speci�c technology changes. The GreenParadox says that investing in alternative-energy technology will increasefossil-fuel use in the short run, which is true also in this model. Ifinvestment in that technology crowds out investments in other types oftechnology, this crowding out can amplify or dampen the Green Paradoxe�ect depending on which type of technology is crowded out. If laborintensive technology AL is crowded out, this will amplify the e�ect of the


future increase in AS. If, instead, AY or AE are crowded out, this willdampen the e�ect of the change in AS. If it crowds out investment inAB, this can either dampen or amplify the e�ect of the future change inAS. To understand the e�ect of crowding out research on AB, considera situation where increased spending on research on alternative-energytechnology crowds out spending on fossil-fuel based technology. This willnot a�ect the current state of technology. When the current researchspending starts to have a signi�cant e�ect on the state of technology,the world may still be in a situation where fossil fuels are the dominantenergy source. In that case the worsening of the state of the fossil-fuelbased technology may decrease the supply of energy enough so that fossilfuel will be reallocated from both the present and the distant future tothe intermediate future. Whether or not this will occur is a quantitativeissue.

Summing up, this section gives the basic results concerning the ef-fect of the future state of technology on fossil-fuel use, as described inproposition 2.2. An improvement in the future state of TFP, energy-saving technology or alternative energy will increase fossil-fuel use inthe short run. An improvement in the future state of the labor aug-menting technology will decrease fossil-fuel use in the short run. Thee�ect of an improvement in the state of technology for fossil-fuel basedenergy generation is ambiguous.

2.3 Two-period model with capital

So far, fossil-fuel use has been the only endogenous variable. In thissection I will also include capital and investments will be endogenouslydetermined. For the general functional form, this complicates the anal-ysis signi�cantly. In this section I will therefore use a two-period modelwhere the endogenous choices are the division of fossil-fuel use betweenthe �rst and second periods and how much, out of �rst-period produc-tion, to invest into second-period capital. In section 2.5 I will insteadassume that θ = σY = 1 but use an in�nite time horizon.

2.3.1 Model setup

Let production depend on fossil-fuel use B, capital K and a vector ofexogenously given variables Γ. Capital will be assumed to be combinedwith labor according to a Cobb-Douglas production function into aninput that is complementary to energy. The exogenous variables in Γwill be the same as above with the exception that the technology factorfor the complementary input (previously AL) will now be called AKL,since it gives the productivity of the combination of capital and labor.

2.3. TWO-PERIOD MODEL WITH CAPITAL 23

The production function is

F (B,K; Γ) = AY

[γKL

(AKLK

αL1−α)σY −1


σY

] σYσY −1

,

(2.18)where YE is the same as before and de�ned in (2.12).

Total fossil-fuel supply is Q and, since there are no externalities, allof it will be used. The initial capital stock, K1, is given and there is fulldepreciation of capital between periods. This gives the following set ofequations:

C1 =F (B1, K1; Γ1)−K2

C2 =F (B2, K2; Γ2)

B2 =Q−B1.

2.3.2 Equilibrium

Since there are no externalities in this model, the planner solution willstill coincide with the competitive equilibrium. The planner problem isto maximize the discounted sum of utility from consumption in the twoperiods. Assuming that fossil-fuel use is strictly positive in both periods(a su�cient condition for this to hold is that σE < ∞) the plannerproblem is

maxB1,K2

U (F (B1, K1; Γ1)−K2) + βU (F (Q−B1, K2; Γ2))

The �rst-order conditions read

B1 :U ′1FB,1 = βU ′2FB,2

K2 :U ′1 = βU ′2FK,2.

These can be rewritten to give the optimality conditions

FB,2 =FK,2FB,1 (2.19)

U ′1 = βU ′2FK,2, (2.20)

where the �rst condition is the Hotelling rule and the second condition isthe Euler equation. The Hotelling rule says that, at the margin, one unitof fossil fuel should contribute equally to second-period production if itis used in production in period 2 or if it is used in production in period1 and the resulting production is invested into second period capital.


2.3.3 Changes in Γ2

For a given Γ2, (2.19) and (2.20) can, in principle, be solved for theequilibrium values of B1 and K2. This means that the system implicitlyde�nes B1 and K2 as functions of Γ2. In order to see how the equilibriumvalues depend on Γ2, consider a change in X which is one of the variablesin Γ2. The change in X will induce endogenous changes in B1 and K2.Let primes denote derivatives with respect to X.

Di�erentiating both sides of (2.19) with respect to X, and notingthat B′2 = −B′1, we obtain

d

dXFB,2 =FB,2

[FBK,2FB,2

K ′2 +FBB,2FB,2

B′2 +FBX,2FB,2

]= {B′2 = −B′1} =

=FB,2

[FBK,2FB,2

K ′2 −FBB,2FB,2

B′1 +FBX,2FB,2

]d

dXFK,2FB,1 =FK,2FB,1

[FKK,2FK,2

K ′2 +FBK,2FK,2

B′2 +FKX,2FK,2

+FBB,1FB,1

B′1

]=FB,2

[FKK,2FK,2

K ′2 +

(FBB,1FB,1

− FBK,2FK,2

)B′1 +

FKX,2FK,2

].

These derivatives must be equal; equating them and rewriting gives

K ′2 =

FBB,2FB,2

− FBK,2FK,2

+FBB,1FB,1

FBK,2FB,2

− FKK,2FK,2

B′1 +

FKX,2FK,2

− FBX,2FB,2

FBK,2FB,2

− FKK,2FK,2

. (2.21)

Similarly, both sides of condition (2.20) can be di�erentiated with respectto X

d

dXU ′1 = = U ′1

U ′′1U ′1

[FB,1B′1 −K ′2] = U ′1θ

1

C1

[K ′2 − FB,1B′1]

d

dXβU ′2FK,2 = βU ′2FK,2

U ′′2U ′2

(FB,2B′2 + FK,2K

′2 + FX,2)

+βU ′2FK,2

[FKK,2FK,2

K ′2 +FBK,2FK,2

B′2 +FKX,2FK,2

]=U ′1

[(FKK,2FK,2

− θFK,2F2

)K ′2 −

(FBK,2FK,2

− θFB,2F2

)B′1

]+U ′1

[FKX,2FK,2

− θFX,2F2

].

Again, these derivatives must be the same; equating them and rewritinggives

K ′2 =θ 1C1FB,1 + θ

FB,2F2− FBK,2

FK,2

θ 1C1

+ θFK,2F2− FKK,2

FK,2

B′1 +

FKX,2FK,2

− θFX,2F2

θ 1C1

+ θFK,2F2− FKK,2

FK,2

. (2.22)

2.3. TWO-PERIOD MODEL WITH CAPITAL 25

Equalizing the right-hand sides of (2.21) and (2.22) and rewriting gives

B′1 =

FKX,2FK,2

−FBX,2FB,2

FBK,2FB,2

−FKK,2FK,2

−FKX,2FK,2

−θFX,2F2

θ 1C1

+θFK,2F2−FKK,2FK,2

θ 1C1FB,1+θ

FB,2F2−FBK,2FK,2

θ 1C1


−FBB,2FB,2

−FBK,2FK,2

+FBB,1FB,1

FBK,2FB,2

−FKK,2FK,2

.

Rearranging the numerator and using that FBKFB

= 1σY

FKF

gives

B′1 =

(FKK,2FK,2

− θFK,2F2

)FBX,2FB,2

−(

1σY− θ)FKX,2F2

+ θ(FBK,2FB,2

− FKK,2FK,2

)FX,2F2

θ 1C1FB,1+θ

FB,2F2−FBK,2FK,2

θ 1C1


+

FBK,2FK,2

−FBB,2FB,2

−FBB,1FB,1

FBK,2FB,2

−FKK,2FK,2

+θ 1C1

(FKX,2FK,2

− FBX,2FB,2

)θ 1C1FB,1+θ

FB,2F2−FBK,2FK,2

θ 1C1


+

FBK,2FK,2

−FBB,2FB,2

−FBB,1FB,1

FBK,2FB,2

−FKK,2FK,2

.

The denominator (which does not contain any derivatives with respectto X and therefore is the same regardless of which variable X represents)can be shown to be positive.1

This means that the sign of B′1 depends on the sign of the two ex-pressions

FKX,2FK,2

− FBX,2FB,2

(2.23)

and (FKK,2FK,2

− θFK,2F2

)FBX,2FB,2

−(

1σY− θ)FKX,2F2

+θ(FBK,2FB,2

− FKK,2FK,2

)FX,2F2. (2.24)

If the expressions have the same sign, then B′1 will also have that sign.Otherwise the sign will depend on the relative strength of the terms.

In table 2.2, expressions with the same signs as expression (2.23) and(2.24) respectively, when di�erentiating with respect to di�erent X, canbe found (see 2.A.2 for the calculations). In the table, GB, GE and GS

are de�ned as in (2.16) while GKL is

GKL = γKL(AKLK

αL1−α)σY −1

σY . (2.25)

1Adding up the two ratios, the denominator (in the denominator) is positive and

so are all terms in the numerator except for −FBK,2

FK,2

(FBK,2

FB,2− FKK,2

FK,2

). The term

FBK,2

FK,2

FKK,2

FK,2is cancelled by an equal term of opposite sign. The other term can be

cancelled by showing thatFBB,2FKK,2

FB,2FK,2− F 2

BK,2

FB,2FK,2> 0.


wrt sign of (2.23) sign of (2.24)AY 0 θ − 1

AKLσY −1σY

(θ − 1

σY

)AE

1−σYσY

1−σYσY

(1 + α (θ − 1))GKL

+ (θ − 1)(

1 + α 1−σYσY

)GE

AB1−σYσY

GB + 1−σEσE

GS

(

(θ − 1)(

1 + α 1−σYσY

)GE

+1−σYσY

(1 + α(θ − 1))GKL

)GB

−

((1 + α(θ − 1))GKL

+(

1 + α 1−σYσY

)GE

)σE−1σE

GS

AS, S

1σY− 1

σE

(θ − 1

σE

)(1− ασY −1

σY

)GE

+(

1σY− 1

σE

)(1 + α (θ − 1))GKL

Table 2.2: Derivative signs

Proposition 2.3. Assume that (2.13), (2.14) and (2.15) hold. ThendB1

dAKL,2≤ 0, dB1

dX≥ 0 for X ∈ {AY,2AE,2AS,2, S2} and the sign of dB1

dAB,2is

ambiguous.

Proof. Follows from inspection of table 2.2.

Thus, qualitatively, the results are the same here as in proposition2.2. When looking at changes in AB, the results are, as in section 2.2.4,ambiguous. As in that case, consider a situation where GKL and GE

are �xed. The sign of the e�ect will depend on the relative sizes of GB

and GS. For both (2.23) and (2.24), the factor in front of GB is positivewhile the factor in front of GS is negative. This means that B′1 will bepositive if GB >> GS and negative if GS >> GB. This is the samequalitative behavior as in section 2.2.4.

In conclusion, the results from section 2.2.4 hold also in this two-period model with capital.

2.4 Introducing climate change

I will now introduce climate change into the model. Climate change isan externality that comes from the amount of greenhouse gases buildingup in the atmosphere as a result of, among other things, the burningof fossil fuels. Introducing climate change related externalities into themodel will allow me to look at a number of di�erent issues. When thereare externalities in the model, the decentralized equilibrium and theplanner solution will no longer coincide. In a decentralized equilibrium,the externalities are not internalized in agents' decisions, but there will

2.4. INTRODUCING CLIMATE CHANGE 27

still be general equilibrium e�ects that will be taken into account. Inthe planner solution, the climate e�ects will be fully internalized. Fur-thermore, as long as all fossil fuel is used in the planner solution, it canbe implemented as a decentralized equilibrium with taxation. In thissection I will consider a model with in�nite time horizon, but withoutcapital. I will then look at how the allocation of fossil-fuel use dependson the sequence {Γt}∞t=0 both in the decentralized equilibrium and in theplanner solution. Furthermore, I will comment on how the optimal taxrate and the welfare gains from implementing the optimal tax schemedepend on {Γt}∞t=0.

2.4.1 Modeling climate change

Climate change is driven by the concentration of greenhouse gases. Car-bon dioxide, CO2, that is emitted from the burning of fossil fuels is animportant greenhouse gas. As the concentration of greenhouse gasesincreases in the atmosphere, the radiative balance between earth andthe surrounding space changes. Primarily, the amount of radiation leav-ing the earth's atmosphere decreases. This leads to an increase in thetemperature.

The full process of climate change is very complicated. Here I willassume that the climate state can be described as the concentration ofgreenhouse gases in the atmosphere. What primarily matters for the cli-mate is the temperature in the atmosphere. For a given concentration ofgreenhouse gases, the atmospheric temperature will reach a steady staterelatively quickly. Thus, if a time period is long enough, assuming thatthe atmospheric temperature is in steady state is not a bad assumption.This modeling abstracts the temperature changes in the oceans, whichoccur much more slowly.

Carbon that is emitted, in the form of CO2, will partly be absorbedeither in the biosphere, through photosynthesis, or into the oceans. Theprocesses through which carbon moves between di�erent reservoirs arecollectively referred to as the carbon cycle. I will, for most of the de-rived results, assume that the carbon cycle is linear. This assumptionis also made by Nordhaus in the DICE/RICE models. There are somereasons why this assumption may not be ful�lled. The oceans couldbecome saturated, meaning that their capacity to take up carbon fromthe atmosphere decreases, and the increased temperature could alter thebehavior of the biosphere so that it becomes a source, rather than a sink,of carbon in the future. These things aside, I do believe that the modelof the climate that I use here serves as a useful approximation. It can,for instance, be shown that a model of the type used here can replicatethe behavior of the climate system in the DICE/RICE models quite well


(see Golosov et al., 2011).I will �rst describe how I model the carbon cycle and then describe

how I assume that the climate state translates into e�ects on the econ-omy through the damage function.

The carbon cycle

Let Mt be the climate state in period t: the stock of greenhouse gasesin the atmosphere. I will assume that this stock, in period t, movestowards a steady-state value of Mt. This steady-state value will dependon past emissions. The steady-state level could depend in many wayson the entire history of emissions. I will assume that it depends on thetotal amount of past emissions. This means that M0 is given and thatfor t > 0, Mt = M

(∑t−1t′=0Bt′

). The steady-state stock is assumed to

be (weakly) increasing in the total amount of past emissions, that isM ′ (∑t−1

t′=0Bt′)≥ 0. I will further assume that the movement towards

the steady-state is linear. This gives the law of motion for the stock ofgreenhouse gases as

Mt+1 = Mt +Bt − δ

[Mt − M

(t−1∑t′=0

Bt′

)], (2.26)

where δ is the rate at which carbon is removed from the atmospherethrough the carbon cycle.2

De�ning Mt = M(∑t−1

t′=0Bt′)and taking M0 as given, gives that the

climate state in period t+ 1 is

Mt+1 = (1− δ)t+1M0 +t∑

t′=0

(1− δ)t−t′(Bt′ + δMt′

).

Thus, fossil-fuel use in period T a�ects climate from period T + 1 andonwards. When considering the e�ects of changes in the pattern offossil-fuel use, induced by changes in the exogenously given variables,the derivative of the climate state with respect to emissions is needed.This derivative is

dMt

dBT

=

{0 if t ≤ T

(1− δ)t−1−T + δ∑t−1

u=T (1− δ)t−1−u dMu

dBTif t > T

,

the latter of which is positive.

2Here, the climate state evolves according to a law of motion for that state variable.An alternative description would be to let the climate state depend on the historyof fossil-fuel use. What matters for most of the results below is that the e�ect offossil-fuel use in period t1 on the climate state in period t2 > t1 only depends ont2 − t1.


A further assumption that I will make for some of the analysis is thatM is linear

M

(t−1∑t′=0

Bt′

)= M0 +m

t−1∑t′=0

Bt′ . (2.27)

This corresponds to a fraction m of emissions staying in the atmosphereforever. With this shape of M , for t > 0 we have

Mt = (1− δ)tM0 +t−1∑t′=0

((1− δ)t−1−t′ + δm

t−2−t′∑u=0

(1− δ)u)Bt′

+δt−1∑t′=0

(1− δ)t′M0

=t−1∑t′=0

(m+ (1−m)(1− δ)t−1−t′

)Bt′

+M0 + (1− δ)t(M0 − M0

). (2.28)

Two relatively similar special cases are δ = 0 and m = 1. Settingδ = 0 gives

Mt = M0 +t−1∑t′=0

Bt′ . (2.29)

When δ = 0, no carbon is taken up in the carbon cycle. Thereforeemissions stay in the atmosphere forever and all emissions add to theclimate state.

Setting m = 1 instead, we obtain

Mt = M0 + (1− δ)t(M0 − M0) +t−1∑t′=0

Bt′ . (2.30)

Here, all emissions still stay forever in the atmosphere since they addto the equilibrium concentration. In addition, any di�erence betweenactual and equilibrium concentration in period 0 decays over time. IfM0 = M0 the two cases are equivalent.

Recent research (see Archer, 2005) indicates that a signi�cant shareof current emissions will stay in the atmosphere for a signi�cant time,so making one of these assumptions may not be a bad approximation.

Under the linearity assumption of M , the derivative of the climatestate with respect to past fossil-fuel use is

dMt

dBT

=

{0 if t ≤ Tm+ (1−m)(1− δ)t−1−T if t > T

. (2.31)


For both of the cases δ = 0 and m = 1, dMt

dBT= 1 for t > T .

Thus, if M is assumed to be linear and given by (2.27), the climatestate in period t is given by (2.28) and the e�ect of fossil-fuel use inperiod T on the climate state in period t is given by (2.31).

If, furthermore, either

δ = 0 or m = 1 and M0 ≤ M0, (2.32)

thent2 ≥ t1 ⇒Mt2 ≥Mt1 (2.33)

anddMt

dBT

=

{0 if t ≤ T1 if t > T

. (2.34)

The damage function

The climate can a�ect the economy in a number of di�erent ways. Here,I will, as in the DICE/RICE-models and in Golosov et al. (2011), assumethat climate a�ects the economy as a multiplicative factor on production.That is, climate change a�ects total factor productivity.

Consumption is equal to production:

Ct = Yt = D(Mt)F (Bt; Γt), (2.35)

where D(M) is the damage function. I will assume that D(M) ∈ (0, 1]and that D′(M) < 0. When it comes to the second derivative, it isnot obvious what the sign should be. In the DICE and RICE models,damages are a concave function of the temperature, while temperature isa convex (logarithmic) function of the concentration of greenhouse gasesin the atmosphere.

I will often use the assumption that

d

dM

D′(M)

D(M)≤ 0. (2.36)

Examples of damage functions that ful�ll this assumption includes allconcave functions (D′′(M) ≤ 0) and D(M) = e−κM .

Under this assumption.

d

dM

D′(M)

D(M)=D′′(M)

D(M)− D′(M)2

D(M)2=D′(M)

D(M)

(D′′(M)

D′(M)− D′(M)

D(M)

)≤ 0

⇒ D′′(M)

D′(M)− D′(M)

D(M)≥ 0

Thus, if the damage function ful�lls (2.36) and θ ≥ 1 then


Mt2 ≥Mt1 ⇒D′(Mt1)

D(Mt1)− D′(Mt2)

D(Mt2)≥ 0 (2.37)

and

D′′(M)

D′(M)− D′(M)

D(M)≥ 0 and

D′′(M)

D′(M)− θD

′(M)

D(M)≥ 0. (2.38)

For some derivations I will make the more speci�c assumption

D(M) = e−κM ; (2.39)

with this damage function

D′(M)

D(M)=D′′(M)

D′(M)= −κ (2.40)

This completes the description of how climate change enters the model.As emphasized above, when climate change is included as an exter-

nality in the model, the planner solution and the competitive equilibriumwill no longer coincide and they need to be solved for separately. Thisis the subject of the following sections.

2.4.2 Decentralized equilibrium

I will now specify and solve for a competitive equilibrium. I will consideran equilibrium with a tax on sales of fossil fuel. For most of this section,I will treat taxes as given. In section 2.4.4 I will then look at the taxesthat implement the optimal solution.

To begin with, I will specify who receives what income and whomakes which decisions. I will assume that households derive income fromsupplying labor and from pro�ts in the energy supplying �rms (given aconstant returns to scale production technology, the �nal good producing�rms will not make any pro�ts). Furthermore, the government's taxrevenues are paid back as lump-sum amounts to the households. Let gtbe the lump-sum transfer in period t.

Without any assumed disutility from work, labor will be inelasticallysupplied in quantity L. The energy supplying �rms are assumed to beowned in equal shares by all households. The households can also savein the form of a riskless bond. Let πB,t and πS,t be the pro�ts from fossil-fuel and alternative-energy supplying �rms, respectively. Let 1

rt+1be the

price in period t of a riskless bond that pays one unit of consumptionin period t + 1 and let at be the holding of riskless bonds that pays inperiod t.


The budget constraint of the households is

Ct +1

rt+1

at+1 = πB,t + πS,t + wt + lt + gt + at. (2.41)

A competitive equilibrium with taxation consists of sequences ofquantities {Bt, Ct}∞t=0, fossil-fuel taxes {τt}∞t=0, lump-sum tax rebates{gt}∞t=0 and prices {pB,t, pS,t, rt+1, wt}∞t=0 such that

• households maximize their total utility∑∞

t=0 βtU(Ct) subject to

the budget constraint (2.41)

• the bond market clears: at = 0 for all t

• prices are competitively determined

• fossil-fuel extracting �rms maximize the discounted sum of pro�tsfrom extraction

• the government balances its budget in each period

Solving for the decentralized equilibrium

Households The utility maximization problem of the representativehousehold is

max{Ct,at+1}∞t=0

∞∑t=0

βtU(Ct)

s.t. Ct +1

rt+1

at+1 = wtL+ at + πB,t + πS,t + gt ∀t.

The Lagrangian of the problem is

L =∞∑t=0

(βtU(Ct) + λt

[wtL+ at + πB,t + πS,t + gt − Ct −

1

rt+1

at+1

]).


Ct : βtU ′(Ct) = λt

at+1 :λt1

rt+1

= λt+1.

Combining these conditions determines the price of the riskless bond:

rt+1 =1

β

U ′(Ct)

U ′(Ct+1). (2.42)

Market clearing in the bond market (at = 0) gives that

Ct = wtLt + πB,t + πS,t + gt. (2.43)


Competitive prices Assuming perfect competition, the prices of theinputs are given by their respective marginal products

pB,t = D(Mt)FB,t, pS,t = D(Mt)FS,t and wt = D(Mt)FL,t. (2.44)

Balanced government budget The government tax revenues, thatare paid back as lump sums to the households, are

gt = τtpB,tBt. (2.45)

Fossil-fuel extracting �rms The fossil-fuel suppliers are assumed tomaximize discounted pro�ts from extraction over time. The discountrate used between pro�ts in periods t and t + 1 is rt+1. Since there areno extraction costs, the pro�t that the �rm makes, per extracted unitof fossil fuel, in period t is (1− τt)pB,t. The maximization problem of afossil-fuel extracting �rm, that has a fossil-fuel resource with quantity q,is

max{bt}∞t=0

∞∑t=0

[t−1∏t′=0

1

rt′+1

](1− τt)pB,tbt s.t.

∞∑t=0

bt ≤ q.


L =∞∑t=0

[t−1∏t′=0

1

rt′+1

]pB,tbt + λ

[q −

∞∑t=0

bt

]+ µtbt.

The �rst-order condition with respect to bt reads[t−1∏t′=0

1

rt′+1

](1− τt)pB,t = λ− µt. (2.46)

Using (2.42), we obtain

t−1∏τ=0

1

rτ+1

= βU ′(C1)

U ′(C0)βU ′(C2)

U ′(C1). . . β

U ′(Ct)

U ′(Ct−1)= βt

U ′(Ct)

U ′(C0).

For any t and T such that bt > 0 and bT > 0, this implies that

βtU ′(Ct)

U ′(C0)(1− τt)pB,t = βT

U ′(CT )

U ′(C0)(1− τT )pB,T .

orβtU ′(Ct)(1− τt)pB,t = βTU ′(CT )(1− τT )pB,T . (2.47)

At the aggregate level, this condition must hold between any periods tand T such that Bt > 0 and BT > 0.


Equilibrium conditions Without any extraction costs for fossil fuel,or the alternative-energy inputs, the pro�ts that the households receivein period t are

πB,t = (1− τt)pB,tBt

πS,t = pS,tSt

Substituting these expressions, and the lump-sum transfers from (2.45),into the households' budget constraint (2.43) and using the fact that theproduction function has constant returns to scale in L, B, and S, we seethat

Ct =D(Mt)FL,tLt + (1− τt)D(Mt)FB,tBt

+D(Mt)FS,tSt + τtD(Mt)FB,tBt

=D(Mt) (FL,tLt + FB,tBt + FS,tSt)

=D(Mt)F (Bt; Γt) = Yt.

Thus, consumption is equal to production. Substituting this quantityinto the Hotelling rule (2.47) gives

βtU ′(Yt)(1− τt)pB,t = βTU ′(YT )(1− τT )pB,T .

Using the de�nition of H from (2.5), the Hotelling rule can be written

βt(1− τt)Ht = βT (1− τT )HT , (2.48)

where Ht = H(Bt; Γt).All fossil fuel will always be extracted in a decentralized equilibrium

with competitive fossil-fuel supply since, as long as the fossil-fuel priceis strictly positive, increased extraction in any period increases pro�ts.

In sum, the decentralized equilibrium is characterized by the Hotellingrule, (2.48) and

∑∞t=0Bt = Q0.

So far the equilibrium looks exactly like the equilibrium without ex-ternalities. This is because the decisions do not internalize the climatee�ects. However, the factor D(Mt) is endogenously determined and willa�ect the equilibrium fossil-fuel use. This will become apparent in thenext section.

Changes in {Γt}∞t=0

I will now consider changes in XT , which is one of the variables in ΓT ,for some T > 0 such that BT > 0. This change will induce endogenouschanges in {Bt}∞t=0. When XT changes, the equilibrium conditions willstill be satis�ed. Thus, when di�erentiating with respect to XT , the


derivative of both sides of the Hotelling rule (2.48) must must be thesame for all t such that Bt > 0.

Let the full derivatives of Ht, Bt, Mt and Ct with respect to XT bedenoted by primes. I will show that, in some speci�ed cases, B′t will havethe same sign for all t < T and that if B′T does not have the oppositesign, then B′t will have the same sign for all t which would violate theconstraint on the total amount of available fossil fuel. The di�erence,compared to section 2.2.4, is that changes in fossil-fuel use in one periodwill change the damages, and therefore potentially the value of fossil-fueluse, in all future periods.

I will start by di�erentiating the Hotelling rule to get a relationshipbetween changes in Bt in di�erent periods. I will then show what theserelationships lead to in two special cases.

With Ht given by (2.5), the derivative of Ht with respect to XT is

H ′t =

Ht

[U ′′tU ′tC ′t +

D′tDtM ′

t +FBB,tFB,t

B′t

]if t 6= T

HT

[U ′′TU ′TC ′T +

D′TDTM ′

T +FBB,TFB,T

B′T +FBX,TFB,T

]if t = T

,

where the derivative of consumption is

C ′t =d

dXt

D(Mt)Ft =

Ct[D′tDtM ′

t +FB,tFtB′t

]if t 6= T

CT

[D′TDTM ′

T +FB,TFT

B′T +FX,TFT

]if t = T

.

Using the form of the utility function (2.10), we obtain

U ′′

U ′C = −θ.

Also, to simplify the notation, de�ne

ξB,t = θFB,tFt− FBB,t

FB,t> 0 and ξX,t =

FBX,tFB,t

− θFX,tFt

. (2.49)

Note that ξX has the same sign as (2.17), the expression that determinesthe sign of the e�ects in the model without externality in proposition2.2.

Using this notation, substituting C ′t in H′t and rewriting gives

H ′t =

Ht

[(1− θ)D

′t

DtM ′

t − ξB,tB′t]

if t 6= T

HT

[(1− θ)D

′T

DTM ′

T − ξB,TB′T + ξX,T

]if t = T

.

The �rst two terms relate to the e�ects of the endogenous changes in{Bt}. The �rst term captures the change in the externality. If Mt in-creases, this decreases production in period t and decreases the marginal


product of fossil fuel. Which of these e�ects dominates, and consequentlywhether the marginal value of fossil-fuel use increases or decreases, de-pends on the value of θ. In the second term, the factor in front of B′t isnegative, capturing that the value of fossil-fuel use decreases as the useincreases. The third term in the expression for t = T captures the directe�ect of the change in XT on HT . The e�ect consists of an e�ect on themarginal product of fossil fuel and an e�ect on production that a�ectsthe marginal value of consumption.

The e�ects of the change in XT can be summarized in the conditions

βt1(1− τt1)H ′t1 = βt2(1− τt2)H ′t2 (2.50)

for all t1 and t2 such that Bt1 > 0 and Bt2 > 0 and

∞∑t=0

B′t = 0.

From the Hotelling rule (2.48), βt1(1−τt1)Ht1 = βt2(1−τt2)Ht2 . Thecomparison between the changes in the two time periods will depend onwhether one of them is T or not.

Consider �rst the case where t1 6= T and t2 6= T . Then

(1− θ)D′t1Dt1

M ′t1− ξB,t1B′t1 = (1− θ)

D′t2Dt2

M ′t2− ξB,t2B′t2 .

Solving for B′t2 gives

B′t2 =ξB,t1ξB,t2

B′t1 + (1− θ)D′t2Dt2

M ′t2− D′t1

Dt1M ′

t1

ξB,t2. (2.51)

Consider now the case t1 = t and t2 = T . Then

(1− θ)D′t

Dt

M ′t − ξB,tB′t = (1− θ)D

′T

DT

M ′T − ξB,TB′T + ξX,T .

Solving for B′T gives

B′T =ξB,tξB,T

B′t + (1− θ)D′TDTM ′

T −D′tDtM ′

t

ξB,T+ξX,TξB,T

. (2.52)

These conditions can now be used to see how the pattern of fossil-fuel use changes in response to the change in XT . We will �rst look atlogarithmic utility and then consider higher curvature.


The log-utility case

Assume �rst that utility is logarithmic: θ = 1. The second terms in(2.51) and (2.52) will then be zero. If considering the case t1 6= T andt2 6= T , then (2.51) implies that


B′t1 . (2.53)

Thus, B′t1 and B′t2must have the same sign. This leads to the following

proposition

Proposition 2.4. If θ = 1, then a change in XT will have the followinge�ect on fossil-fuel use

Sgn (B′t) = Sgn

(FX,TFT− FBX,T

FB,T

)for any t 6= T such that Bt > 0

and

Sgn (B′T ) = Sgn

(FBX,TFB,T

− FX,TFT

).

Proof. From (2.53) it follows that B′t must have the same sign for allt 6= T . Since

∑∞t=0 B

′t = 0, B′T must have the opposite sign. From

(2.52), with θ = 1, it follows that B′T can only have the opposite sign toB′t if B

′T has the same sign as ξX,T . The proposition then follows from

the de�nition of ξX,T , (2.49).

Note that this proposition has the same implications as proposition2.2. This is because when utility is assumed to be logarithmic, theexternalities disappear from the decentralized equilibrium and thereforethe e�ects are the same as in the model without externalities. Thisindependence of externalities can be interpreted in two di�erent ways.The �rst interpretation is that the e�ects on the marginal productivityof fossil fuel and the marginal utility of consumption exactly cancel eachother out when utility is logarithmic. The second interpretation is thatif utility is logarithmic, then the externality enters as an additive termin the utility function. A di�erent way to make the damages irrelevantfor the decentralized equilibrium is to assume that damages enters as anadditive term in the utility function instead of a�ecting productivity. Ifthis assumption is made, the externalities will not a�ect the decentralizedequilibrium and consequently the e�ect will be the same as in the modelwithout externalities regardless of the value of θ.


The case θ > 1 and δ = 0 or m = 1 and M0 ≤ M0

When θ 6= 1, the damages do not cancel from conditions (2.51) and(2.52) since the e�ects on marginal product of fossil fuel and on marginalutility of consumption no longer cancel. If θ > 1, the e�ect on marginalutility of consumption dominates the e�ect on marginal productivity offossil fuel and higher damages increase the value of fossil-fuel use. Theopposite holds if θ < 1. When the e�ects on the climate state mustbe taken into account, the analysis of the e�ects of a change in XT onfossil-fuel use becomes, in general, signi�cantly more complicated sincethe e�ects of changes in fossil-fuel use in some period on the damagesin all future periods must be taken into account. In this section, I willconsider the case θ > 1 and show that if the damage function ful�lls(2.36) and the carbon cycle ful�lls (2.32), that is, either δ = 0 or m = 1and M0 ≤ M0, then the sign of B′t will be the same for all t < T and B′Tmust have the opposite sign.

Consider a change in XT and a time period t2 > 0 such that t2 6= T ,Bt2 > 0 and such that B′t has the same sign for all t < t2. For this case(2.51) gives that


B′t1 + (θ − 1)

D′t1Dt1

M ′t1− D′t2

Dt2M ′

t2

ξB,t2(2.54)

for some t1 < t2 such that Bt1 > 0 and t1 6= T . The �rst term, asbefore, has the same sign as B′t1 . The second term captures the changein relative pro�tability of using fossil fuel in the two periods that comesfrom the changes in damages caused by the change in fossil-fuel use.When θ > 1, the second term has the same sign as

D′t1Dt1

M ′t1−D′t2Dt2

M ′t2.

Under assumption (2.32)

M ′t =

t−1∑t′=0

B′t′

giving that

M ′t2

= M ′t1

+

t2−1∑t=t1

B′t

implying that

D′t1Dt1

M ′t1−D′t2Dt2

M ′t2

=D′t1Dt1

M ′t1−D′t2Dt2

(M ′

t1+

t2−1∑t=t1

B′t

).


Substituting this into (2.54) delivers

B′t2 =ξBt1ξBt2

B′t1 +θ − 1

ξB,t2

[(D′t1Dt1

−D′t2Dt2

)M ′

t1−D′t2Dt2

t2−1∑t=t1

B′t

].

Under assumption (2.32), Mt2 ≥ Mt1 . Since the damage function isassumed to ful�ll (2.36), (2.37) implies that the factor in front of M ′

t1is

positive. By assumption, B′t1 , M′t1and B′t all have the same sign. Since

the factors in front of each of them are positive, B′t2 must also have thatsame sign.

This leads to the following proposition:

Proposition 2.5. If θ > 1, the damage function ful�lls (2.36) and thecarbon cycle ful�lls (2.32), then the e�ects of a change in XT will besuch that

Sgn (B′t) = Sgn

((θFX,TFT− FBX,T

FB,T

))for any t < T such that Bt > 0

and

Sgn (B′T ) = Sgn

((FBX,TFB,T

− θFX,TFT

))Proof. From the argument above, it follows that B′t will have the samesign for all t < T . If BT had the same sign then B′t would have thesame sign for all t which would violate the condition that

∑∞t=0B

′t = 0.

This means that B′T must have the opposite sign compared to B′t forany t < T . From (2.52) it follows that this change in sign must comefrom ξX,T and therefore that B′T must have the same sign as ξX,T . Theproposition then follows from the de�nition of ξX,T in (2.49).

Compared to the case without externalities, it is more di�cult topredict the e�ects for t > T since then the changes in damages enter thecomparisons.

In conclusion, proposition 2.5 again con�rms the results of section2.2.4.

By closer inspection of the signs of B′t for t ≤ T , it is clear thatproposition 2.5 should hold under much more general conditions. Thedetails of this argument, however, is not outlined here. What is requiredis that a su�cient amount of CO2 stays in the atmosphere for a su�cientamount of time.


2.4.3 Planner solution

The section above dealt with e�ects of introducing externalities in thedecentralized equilibrium. Since the climate e�ects are not internalizeddirectly in any decisions made, the e�ects on the equilibrium allocationcame through indirect e�ects. In the planner solution, the e�ects offossil-fuel use on the climate are taken into account directly. One impli-cation of this is that it is no longer certain that the planner will chooseto use all the fossil fuel. For much of the analysis I will, however, focuson the case where the constraint on total fossil-fuel supply does bind.

Characterizing the planner solution

The planner wants to choose fossil-fuel use to maximize the discountedsum of utility for the representative household in the economy. Con-sumption is, in each period, given by (2.35) and the law of motion ofthe climate state is (2.26). Furthermore, the planner is constrained bythe total supply of fossil fuel. As in the decentralized equilibrium, I willstill assume that {Γt}∞t=0 is an exogenously given sequence. This givesthe following formulation of the planner problem:

max{Bt}∞t=0

∞∑t=0

βtU(Ct)

s.t Ct = D(Mt)F (Bt; Γt)∞∑t=0

Bt ≤ Q0

Mt+1 = Mt +Bt − δ

[Mt − M

(t−1∑t′=0

Bt′

)]Bt ≥ 0.

The Lagrangian of this problem can be written

L =∞∑t=0

βtU (D(Mt)F (Bt; Γt)) + λ

[Q0 −

∞∑t=0

Bt

]+ µtBt,

where λ is positive if the constraint on total available fossil fuel bindsand µt is positive if the non-negativity constraint on fossil-fuel use bindsin period t. While not written out explicitly here, the law of motion ofthe climate state makes Mt a function of B0, B1, . . . , Bt−1.

Taking the �rst order condition with respect to Bt gives

λ− µt = βtU ′tDtFB,t +∞∑

t′=t+1

βt′U ′t′Ft′D

′t′dMt′

dBt

.


The �rst term on the right hand side is the marginal discounted utilityof the marginal product of fossil-fuel use in period t. The sum is thetotal discounted marginal utility e�ects, in all future periods, caused byemissions in period t.

To simplify notation, let

H1,t = U ′tDtFB,t and H2,t,t′ = βt′U ′t′Ft′D

′t′dMt′

dBt

. (2.55)

Here H1,t is the same as Ht, de�ned in (2.5).The only di�erence between H2,t,t′ for di�erent values of t comes from

the factor dMt′dBt

. If M is assumed to be linear and given by (2.27), thendMt′dBt

is given by (2.31) and is therefore a constant that depends only onthe di�erence between t and t′.

Using (2.55), the optimality condition can be written

βtH1,t +∞∑

t′=t+1

H2,t,t′ = λ− µt. (2.56)

When the constraint on total amount of available fossil fuel does notbind, λ = 0 and the solution is characterized by

βtH1,t +∞∑

t′=t+1

H2,t,t′ = 0 (2.57)

for all t such that Bt > 0.When the constraint binds, the solution is characterized by (2.56)

and∑∞

t=0Bt = Q0. Consider t1<t2 such that Bt1 > 0 and Bt2 > 0.Then

βt1H1,t1 +∞∑

t′=t1+1

H2,t1,t′ = βt2H1,t2 +∞∑

t′=t2+1

H2,t2,t′ . (2.58)

This equation says that for any two periods where fossil-fuel use is pos-itive, the marginal value of fossil-fuel use, net of future damages causedby emissions, should be the same.

Changes in {Γt}∞t=0

Consider again a change in XT which is one of the exogenous variablesin ΓT . This change will induce changes in the sequence of fossil-fueluse {Bt}∞t=0. Letting primes denote total derivatives with respect to XT

(taking the response of fossil-fuel use into account), the terms in theoptimality conditions (2.55) will be a�ected as follows:

H ′1,t =

H1,t

[U ′′tU ′tC ′t +

D′tDtM ′

t +FBB,tFB,t

B′t

]if t 6= T

H1,T

[U ′′TU ′TC ′T +

D′TDTM ′

T +FBB,TFB,T

B′T +FBX,TFB,T

]if t = T


and

H ′2,t,t′ =

H2,t,t′

[U ′′t′

U ′t′C ′t′ +

FB,t′

Ft′B′t′ +

D′′t′

D′t′M ′

t′

]if t′ 6= T

H2,t,T

[U ′′TU ′TC ′T +

FB,TFT

B′T +D′′TD′TM ′

T +FX,TFT

]if t′ = T

.

As in the decentralized equilibrium, the change in consumption is

C ′t =

Ct[FB,tFtB′t +

D′tDtM ′

t

]if t 6= T

CT

[FB,TFT

B′T +D′TDTM ′

T +FX,TFT

]if t = T

.

The shape of the utility function (2.10) implies that U ′′tU ′tCt = −θ.

Using this, and the notation de�ned in (2.49), the changes in H1,t andH2,t,t′ can be rewritten as

H ′1,t =

H1,t

[(1− θ)D

′t

DtM ′

t − ξB,tB′t]

if t 6= T

H1,T

[(1− θ)D

′T

DTM ′

T − ξB,TB′T + ξX,T

]if t = T

(2.59)

and

H ′2,t,t′ =

H2,t,t′

[(1− θ)FB,t′

Ft′B′t′ +

(D′′t′

D′t′− θD

′t′

Dt′

)M ′

t′

]if t′ 6= T

H2,t,T

[(1− θ)

(FB,TFT

B′T +FX,TFT

)+(D′′TD′T− θD

′T

DT

)M ′

T

]if t′ = T

.

(2.60)The e�ect of a change in XT depends on whether the constraint on

the total amount of available fossil fuel binds or not. If the constraintdoes not bind, then (2.57) gives that the change in the planner solu-tion's allocation of fossil-fuel use in response to the change in XT can bedescribed by

βtH ′1,t +∞∑

t′=t+1

H ′2,t,t′ = 0

for any t such that Bt > 0. Note that, since the emissions in period t willa�ect the climate state in all future periods, there can still be nontrivialinteractions between the changes of Bt in di�erent periods.

Assume now that the constraint on the total amount of availablefossil fuel binds. Using (2.58), the e�ects of changing XT can then bedescribed by the equations

βt1H ′1,t1 +∞∑

t=t1+1

H ′2,t1,t = βt2H ′1,t2 +∞∑

t=t2+1

H ′2,t2,t


for any t1 < t2 such that Bt1 > 0 and Bt2 > 0 and

∞∑t=0

B′t = 0.

The term

H ′2,t,t′

H2,t,t′=

(1− θ)FB,t′Ft′

B′t′ +(D′′t′

D′t′− θD

′t′

Dt′

)M ′

t′ if t′ 6= T

(1− θ)(FB,TFT

B′T +FX,TFT

)+(D′′TD′T− θD

′T

DT

)M ′

T if t′ = T

(2.61)depends on t′ but is independent of t. The di�erence between H ′2,t1,t′and H ′2,t2,t′ can then be written

H ′2,t1,t′ −H′2,t2,t′ =H2,t1,t′

H ′2,t1,t′

H2,t1,t′−H2,t2,t′

H ′2,t2,t′

H2,t2,t′

= (H2,t1,t′ −H2,t2,t′)H ′2,t1,t′

H2,t1,t′

= βt′U ′t′Ft′D

′t′

(dMt′

dBt1

− dMt′

dBt2

)H ′2,t1,t′

H2,t1,t′.

It follows that

βt2H ′1,t2 − βt1H ′1,t1 =

t2∑t=t1+1

H ′2,t1,t +∞∑

t=t2+1

(H ′2,t1,t −H

′2,t2,t

)=

t2∑t=t1+1

H ′2,t1,t +∞∑

t=t2+1

(H2,t1,t −H2,t2,t)H ′2,t1,tH2,t1,t

=

t2∑t=t1+1

H ′2,t1,t

+∞∑

t=t2+1

βtU ′tFtD′t

H ′2,t1,tH2,t1,t

(dMt

dBt1

− dMt

dBt2

)(2.62)

whereH′

2,t1,t′

H2,t1,t′is given by (2.61).

I will now show what these conditions imply in some special cases.

The log-utility case

Assume �rst that utility is logarithmic: θ = 1.Then (2.55) becomes

H1,t =FB,tFt

> 0 and H2,t,t′ = βt′D′t′

Dt′

dMt′

dBt

< 0. (2.63)


Equation (2.59) gives

H ′1,t =

{−H1,tξB,tB

′t if t 6= T

H1,T [−ξB,TB′T + ξX,T ] if t = T(2.64)

while equation (2.60) delivers

H ′2,t,t′ = H2,t,t′

(D′′t′

D′t′− D′t′

Dt′

)M ′

t′ . (2.65)

Condition (2.62) then becomes

βt2H1,t2ξB,t2B′t2

= βt1H1,t1ξB,t1B′t1

−∞∑

t=t2+1

βtD′tDt

(D′′tD′t− D′tDt

)M ′

t

(dMt

dBt1

− dMt

dBt2

)

−t2∑

t=t1+1

H2,t1,t


)M ′

t (2.66)

if t1 6= T and t2 6= T . If, instead t1 = t and t2 = T , (2.62) becomes

βTH1,T ξB,TB′T = βtH1,tξB,tB

′t + βTH1,T ξX,T

−∞∑

t′=T+1

βt′D′t′

Dt′

(D′′t′


Dt′

)M ′

t′

(dMt′

dBt

− dMt′

dBT

)

−T∑

t′=t+1

H2,t,t′

(D′′t′


Dt′

)M ′

t′ . (2.67)

In general, this condition depends on the entire future, since emis-sions made in any period has e�ects over the entire future.

If the carbon cycle ful�lls (2.32), (2.34) then implies that all termsof the �rst sum are zero. If instead the damage function is assumed tobe exponential, as in (2.39), (2.40) implies that all terms of both sumsare zero.

If the carbon cycle ful�lls (2.32), (2.66) becomes

βt2H1,t2ξB,t2B′t2

= βt1H1,t1ξB,t1B′t1

+

t2∑t=t1+1

H2,t1,t

(D′tDt

− D′′tD′t

)M ′

t .

Assume further that the damage function ful�lls (2.36) and that B′thas the same sign for all t < t2. Combining H2,t1,t ≤ 0 with (2.38) givesthat all the factors in front of M ′

t′ in the sum will be positive. The


assumption that B′t has the same sign for all t < t2 implies that M ′t will

also have that same sign for all t ≤ t2, since M ′t =

∑t−1t′=0B

′t′ . So the

conclusion is that B′t2 will have that sign as well.If, instead, the damage function is exponential, given by (2.39), (2.66)

becomesH1,t2ξB,t2B

′t2

= H1,t1ξB,t2B′t1

(2.68)

and B′t will have the same sign for all t 6= T .This gives the following proposition:

Proposition 2.6. Assume that utility is logarithmic, θ = 1, and thatthe damage function ful�lls (2.36). Assume further at least one of δ =0, m = 1 or that the damage function is exponential, given by (2.39).Consider a change in XT , which is one of the variables in ΓT . Then theinduced changes in fossil-fuel use are such that

Sgn (B′t) = Sgn

(FX,TFT− FBX,T

FB,T

)for any t < T such that Bt > 0

and

Sgn (B′T ) = Sgn

(FBX,TFB,T

− FX,TFT

)If the damage function is assumed to be exponential, given by (2.39), thesign of B′t will be the same for all t 6= T .

Proof. From the argument above, it follows that B′t must have the samesign for all t 6= T in the exponential case. Since

∑∞t=0 B

′t = 0, B′T must

have the opposite sign. In the case δ = 0 or m = 1, B′t will have thesame sign for all t < T . If BT also had that same sign, B′t would havethe same sign for all t contradicting

∑∞t=0B

′t = 0. From (2.67), with the

assumption δ = 0 or m = 1

βTH1,T ξB,TB′T = βtH1,tξB,TB

′t + βTH1,T ξX,T

−T∑

t′=t+1

H2,t,t′

(D′′t′


Dt′

)M ′

t′

On the RHS the �rst and last terms both have the same sign as B′t forall t < T . It follows that B′T can only have the opposite sign to that ofBt if

Sgn (B′t) = Sgn (−ξX,T ) = Sgn

(FX,TFT− FBX,T

FB,T

)

In conclusion, the conclusions from section 2.2.4 hold here as well.


The case θ > 1 and either δ = 0 or m = 1 and M0 ≥M0

I will now show that for θ > 1, and if the carbon cycle ful�lls (2.32), thechanges in B′t, for all t < T , induced by a change in ΓT , must have thesame sign.

Consider t1 6= T such that Bt1 > 0 and t2 6= T such that t2 is thesmallest t > t1 with Bt > 0. This implies that for any t such thatt1 < t ≤ t2, Mt ≥ Mt1 and M ′

t = M ′t1

+ B′t1 . I also assume that B′t hasthe same sign for all t ≤ t1.

Also, under assumption (2.36), for t > t1, (2.37) gives

Mt ≥Mt1 ⇒D′tDt

≤D′t1Dt1

.

Using (2.34), (2.62) gives

βt2H ′1,t2 − βt1H ′1,t1 =

t2∑t=t1+1

H ′2,t1,t, (2.69)

where

H ′1,t1 =H1,t1

[(1− θ)

D′t1Dt1

M ′t1− ξB,t1B′t1

]H ′1,t2 =H1,t2

[(1− θ)

D′t2Dt2


]=H1,t2

[(1− θ)

D′t2Dt2

(M ′

t1+B′t1

)− ξB,t2B′t2

]H ′2,t1,t =H2,t1,t

[(1− θ)FB,t

FtB′t +

(D′′tD′t− θD

′t

Dt

)M ′

t

].

Under the assumptions made here, B′t = 0 if t1 < t < t2, and M ′t =

M ′t1

+B′t1 , implying

t2∑t=t1+1

H ′2,t1,t =

t2∑t=t1+1

H2,t1,t

[(1− θ)FB,t

FtB′t +


′t

Dt

)M ′

t

]=H2,t1,t2(1− θ)FB,t2

Ft2B′t2

+

t2∑t=t1+1

H2,t1,t


′t

Dt

)(M ′

t1+B′t1

)(2.70)


Substituting (2.70) into (2.69) and rewriting delivers (see appendix 2.A.3)

B′t2 =βt2H1,t2(1− θ)D

′t2

Dt2+ βt1H1,t1ξB,t1 −

∑t2t=t1+1H2,t1,t


′t

Dt

)βt2H1,t2ξB,t2 +H2,t1,t2(1− θ)FB,t2

Ft2

B′t1

+βt2H1,t2(θ − 1)

(D′t1Dt1− D′t2

Dt2

)βt2H1,t2ξB,t2 +H2,t1,t2(1− θ)FB,t2

Ft2

M ′t1

(2.71)

−

∑t2t=t1+1 H2,t1,t

(D′′tD′t− D′t

Dt+ (θ − 1)

(D′t1Dt1− D′t

Dt

))βt2H1,t2ξB,t2 +H2,t1,t2(1− θ)FB,t2

Ft2

M ′t1

Under the assumptions made here, the denominator and all the termsin the numerators are positive. To see this, �rst note that Dt > 0,D′t < 0, H1,t > 0, H2,t1,t < 0, ξB,t > 0 and θ > 1; the result then followsfrom (2.33), (2.37) and (2.38). By assumption, M ′

t1and B′t1 have the

same sign. Since all the factors multiplying these factors in (2.71) arepositive, B′t2 will also have that same sign.

This leads to the following proposition:

Proposition 2.7. Assume that θ > 1 and that the damage functionful�lls (2.36). Assume further that δ = 0 or that m = 1 and M0 = M0.Consider a change in XT which is one of the variables in ΓT . Then B

′t

will have the same sign for all t < T .

Furthermore, consider a situation where ΓT changes in such a waythat the value of using fossil fuel in period T decreases. Could thatlead to an increase in Bt for t < T? I will argue here that this isunlikely. If fossil-fuel use was unaltered, the value of fossil-fuel use in allother periods would be higher than in period T . Assume then that B0

decreased. Proposition 2.7 then implies that Bt would decrease for allt < T . Since the climate state from period T and onwards would thenbe improved, this should, if anything, decrease the value of fossil-fueluse also for t > T .

2.4.4 Optimal taxation

We have seen above that changes in the technology trends will change thepattern of fossil-fuel use over time, both in the socially optimal plannersolution and in a decentralized equilibrium. This also has implicationsfor how taxation should be used to move the competitive equilibriumtowards the planner solution. In this section I will describe the implica-tions for the optimal tax on fossil-fuel use.


In a model without any costs of extracting fossil fuel, the planner so-lution can be implemented in a competitive equilibrium with taxation ifand only if all fossil fuel is exhausted in the planner solution.3 So assumein the following that all fuels are exhausted in the planner solution.

Furthermore, in a model without extraction costs and without anydistributional considerations, the optimal tax system is not uniquely de-termined. This is because the tax system need only a�ect the relativepro�tability of extraction in di�erent periods. Therefore, only the rel-ative tax rate across time periods, and not the level of the tax rate, isdetermined. Here I will look at the particular tax system that exactlyinternalizes the future climate related damages (along the equilibriumpath) in the fossil-fuel price in period t. This tax system is robust tointroducing extraction costs. Choosing di�erent levels of taxes (but stillmaintaining the right intertemporal relation between the tax rate indi�erent periods) would redistribute income to or from the fossil-fuel re-source owners. In this model, the fossil-fuel resources are owned in equalshares by all the households and therefore these distributional concernsdo not matter here.

Writing out the equilibrium condition (2.48), with Ht given by (2.5),gives

βt1U ′t1Dt1FB,t1 (1− τt1) = βt2U ′t2Dt2FB,t2 (1− τt2) . (2.72)

Since this condition only depends on the ratio between 1− τ1 and 1− τ2,the same equilibrium allocation can be supported by any sequence oftaxes that gives the same ratio for any t1 and t2; this is the indeterminacydescribed above.

The optimality condition from the planner solution (2.58) can bewritten

βt1U ′t1Dt1FB,t1+∞∑t=t1

βtU ′tFtD′t

dMt

dBt1

= βt2U ′t2Dt2FB,t2+∞∑t=t2

βtU ′tFtD′t

dMt

dBt2

.

The tax system that exactly internalizes future climate damages issuch that, for the sequence {Bt}∞t=0 that solves the planner problem, andany t such that Bt > 0,

βtU ′tDtFB,t(1− τt) = βtU ′tDtFB,t +∞∑t′=t


′t′dMt′

dBt

.

3If a proportional tax on fossil-fuel use was 100% in all time periods, then anyallocation of fossil-fuel use can be an equilibrium; if the tax rate is lower than thatin any period, then all fuels must be exhausted in the competitive equilibrium.


Simpli�ed, this becomes

DtFB,tτt = −∞∑t′=t

βt′U ′t′

βtU ′tFt′D

′t′dMt′

dBt

. (2.73)

This equation shows that the tax should balance the current value ofusing fossil fuel against the discounted sum of future damages.

The left-hand side of (2.73) is

DtFB,tτt = {(2.44)} = pB,tτt ≡ τt.

This is the per unit tax in period t, in terms of period t consumption(while τt is the tax rate in period t). The right-hand side of (2.73)gives the discounted sum of the value of future marginal damages causedby period t emissions. These are expressed in terms of the value (inutility terms) of lost consumption from future damages. The damagesare then normalized back to period t consumption through division bythe marginal utility of consumption in period t.

Consider now, again, a change in XT which is one of the variablesin ΓT . If this change decreases the value of using fossil fuel in periodT , this should increase the relative value of using fossil fuel in anotherperiod t. This should lead to increased fossil-fuel use in that periodwhich leads to decreased value of using fossil fuel in that period (boththrough decreased marginal productivity and decreased marginal utilityfrom consumption). Assumption (2.36) implies that more emissions in-creases the relative marginal damages. Both of these e�ects suggest thatincreased fossil-fuel use in a period, in the planner solution, implies thatthe optimal tax in that period increases.

Under some speci�c assumptions it can be shown that this intuitionholds. Assume that utility is logarithmic (θ = 1) and that the damagefunction is exponential, as in (2.39). The optimal tax condition thenbecomes

τt =1

βtDtFtκ

∞∑t′=t

βt′ dMt′

dBt

.

If M is assumed to be linear, the sum in the RHS is a constant thatdepends only on m, δ and β. So the per unit tax depends on Dt and Ft.If fossil-fuel use is increased in period t, this will tend to increase thetax. The damage Dt can change in either way depending on all changesin the sequence {Bt′}t−1

t′=0. For t = 0, Dt will be unchanged.Looking instead at the tax rate τt, we have

τt =1

βtFtFB,t

κ

∞∑t′=t

βt′ dMt′

dBt


An increase in Bt increases Ft and decreases FB,t; both changes go inthe direction of increasing the tax rate.

The results of proposition 2.6 give the e�ect on fossil-fuel use ofchanges in XT . This leads to the following proposition.

Proposition 2.8. Assume that θ = 1, D(M) = e−κM , that M is linearand that

∑∞t=0Bt ≤ Q0 binds in the planner solution. Consider changes

in XT , where BT > 0. Then, for any t 6= T such that Bt > 0, thechanges in the optimal tax rate τ and per unit tax τ , are such that

Sgn (τ ′0) = Sgn (τ ′t) = Sgn

((FX,TFT− FBX,T

FB,T

))Proof. Since, τ0 and τt in periods t 6= T , the tax rate changes in thesame direction as fossil-fuel use, the proposition follows from proposition2.6.

In section 2.5 I will show that this result generalizes to a modelwith the same assumptions regarding the utility function and damagefunction with capital as long as capital enters the production functionas in a Cobb-Douglas production function and it depreciates fully.

2.4.5 Welfare e�ects of changing {Γt}∞t=0

Changes in the exogenous variables {Γt}∞t=0 will also have implicationsfor welfare, both in the socially optimal planner solution and in the de-centralized equilibrium. This will also have implications for the welfaregains from imposing the proper tax on fossil-fuel use in the decentral-ized equilibrium. An increase in any variable in Γ in any time periodwill always have a direct positive e�ect by increasing production in thatperiod. In the decentralized equilibrium there will also be indirect e�ectsgoing through the changes in the fossil-fuel use pattern. Since the exter-nalities are internalized in the planner solution, the envelope conditionimplies that there are no such indirect e�ects in the planner solution. Inthis section I will compare the planner solution to the unregulated de-centralized equilibrium and see how changes in {Γt}∞t=0 a�ect the welfaregains from going to the optimal planner solution from the unregulateddecentralized equilibrium. Using a simpli�ed example, I will show thatthe change in the welfare gains from taxation can have either sign.

Let {BPt }∞t=0 and {BD

t }∞t=0 be the fossil-fuel allocation in the plannersolution and the decentralized equilibrium respectively. Let also {CP

t }∞t=0

and {CDt }∞t=0 be the corresponding consumption sequences. Welfare is

then given by V P and V D respectively where

V P =∞∑t=0

βtU(CPt ) and V D =

∞∑t=0

βtU(CDt ).


Letting primes denote derivatives with respect to XT , the welfaree�ects of a change in XT are then

dV

dXT

=∞∑t=0

βtU ′tC′t =

∞∑t=0

βtU ′tCt

[FB,tFt

B′t +D′tDt

M ′t

]+ βTU ′TCT

FX,TFT

=∞∑t=0

βtU ′t [DtFB,tB′t + FtD

′tM′t ] + βTU ′TDTFX,T . (2.74)

The sum represents the indirect e�ects coming from redistributionof fossil-fuel use over time. The second part is the direct e�ect of thechange in XT .

In the planner solution, the indirect e�ects in (2.74) can be rewrittenas

∞∑t=0


′tM′t ]

=∞∑t=0

βtU ′t

[DtFB,tB

′t + FtD

′t

t∑t′=0

dMt

dBt′B′t′

]

=∞∑t=0

βtU ′tDtFB,tB′t +

∞∑t′=0

∞∑t=t′

βtU ′tFtD′t

dMt

dBt′B′t′

=∞∑t=0

βtU ′tDtFB,tB′t +

∞∑t=0

∞∑t′=t+1


′t′dMt′

dBt

B′t

=∞∑t=0

[βtU ′tDtFB,t +

∞∑t′=t+1


′t′dMt′

dBt

]B′t = {(2.55) and (2.56)}

∞∑t=0

[λ− µt]B′t =∞∑t=0

λB′t −∞∑t=0

µtB′t = λ

∞∑t=0

B′t = 0.

Thus, as expected from the envelope theorem, the indirect e�ect is zeroin the planner solution. This leaves only the direct e�ect

dV P

dXT

= βTU ′TDTFX,T . (2.75)

In the decentralized equilibrium, using (2.5) and (2.48) with τt = 0for all t, the indirect e�ect in (2.74) can be rewritten as

∞∑t=0


′tM′t ] =

∞∑t=0

βtHtB′t +

∞∑t=0

βtU ′tFtD′tM′t .


For any t such that B′t 6= 0, βtHt will be the same. Assuming thatBt > 0, βtHt = β tHt for all t such that B′t 6= 0. This delivers

∞∑t=0

βtHtB′t = β tHt

∞∑t=0

B′t = 0.

The implication of this �nding is that the e�ect of the redistribution offossil-fuel use in the decentralized equilibrium, in response to the changein XT , is zero when considering the e�ects on the marginal product offossil fuel. This is because the e�ect on the marginal value of fossil-fueluse in production is taken into account in the decentralized equilibrium.However, the e�ect on the externality is not taken into account and thate�ect will typically not be zero.

This gives the welfare e�ect of a change in XT in the decentralizedequilibrium as

dV D

dXT

=∞∑t=0

βtU ′tD′tFtM

′t + βTU ′TDTFX,T (2.76)

Since, by assumption, FX,T ≥ 0, the derivative of welfare with respectto XT will be positive in the planner solution. The same thing is notgenerally true in the decentralized equilibrium since there the climatee�ects are not taken into account and therefore the climate e�ects, andthe total welfare change, could go either way.

Comparing the welfare e�ects of changes in the two solutions, thedirect e�ect is present in both of them. If the solutions are not toodi�erent (e.g., if the externality is not too severe), the direct welfaree�ect will be similar in both solutions. If utility is logarithmic, whatmatters is how di�erent FX,T

FTis.

If the direct e�ect is similar in the decentralized equilibrium and theplanner solution, the sign of the change in the di�erence between thewelfare in the solutions, that is, the sign of the change in the welfaregains from taxation, will depend only on the sign of the indirect wel-fare e�ect going through the change in the externalities. Typically, adi�erent measure of welfare comparisons, e.g., equivalent variation inconsumption, is used instead of looking at the change in total welfare.However, under the functional-form assumptions made here the resultsfrom both procedures would be qualitatively the same.4

I will now show, by looking at a simple example, that the indirecte�ect can go either way. The simplifying assumptions I will make areθ = 1, σY = 1, σE → ∞ and D(M) = e−κM . I will also assume thateither m = 1 or δ = 0.

4This is true since∑t β

t ln ((1 + ∆C)Ct) = ln(1+∆C)1−β +

∑t β

t ln(Ct).


When the energy sources are perfect substitutes (σE → ∞), fossil-fuel use will typically only be non-zero in a �nite number of periods. Iwill assume that the solution is such that Bt > 0 if and only if t ≤ t.

I will then look at the indirect e�ect of a change in XT , which isa variable in ΓT for some T ≤ t, on the welfare in the decentralizedequilibrium. Under the assumptions made, the externality part of thewelfare change in the decentralized equilibrium (2.76) becomes

∞∑t=0

βtU ′tD′tFtM

′t =

∞∑t=0

βtD′tDt

M ′t = −κ

∞∑t=0

βtM ′t .

Furthermore, under these assumptions, the production function canbe written

F = AYL1−γE(ABB + ASS)γE

implying that

FB = γEABF

ABB + ASSand FBB = (γE − 1)

ABFBABB + ASS

and that

FBF

= γEAB

ABB + ASSand ξB =

FBF− FBB

FB=

ABABB + ASS

=1

γE

FBF.

For any t1 ≤ t and t2 ≤ t, the equilibrium condition (2.48) becomes

βt1U ′(Ct1)D(Mt1)FB,t1 = βt2U ′(Ct2)D(Mt2)FB,t2

βt1FB,t1Ft1

= βt2FB,t2Ft2

.

Since, by assumption, B0 > 0 this implies that, for any t ≤ t

ξB,0 = βtξB,t ⇒ξB,0ξB,t

= βt.

The conditions for the changes in the decentralized equilibrium (2.51)and (2.52) gives, for any t ≤ t

−ξB,0B′0 =

{−ξB,tB′t if t 6= T−ξB,TB′T + ξX,T if t = T

or

B′t =

{βtB′0 if t 6= T

βTB′0 +ξX,TξB,T

if t = T.


Since the constraint on the total amount of available fossil fuel alwaysbinds in the decentralized equilibrium, it must be that

0 =∞∑t=0

B′t =t∑t=0

βtB′0 +ξX,TξB,T

= B′0

t∑t=0

βt +ξX,TξB,T

= B′01− β t+1

1− β+ξX,TξB,T

.

This deliversB′0 = − 1− β

1− β t+1

ξX,TξB,T

and

B′t =

−βt 1−β

1−βt+1

ξX,TξB,T

if t 6= T and t ≤ t

−βt 1−β1−βt+1

ξX,TξB,T

+ξX,TξB,T

if t = T

0 if t > t

. (2.77)

From (2.31), the change in the climate state is

M ′t =

t−1∑t′=0

dMt

dBt′B′t′ =

t−1∑t′=0

(m+ (1−m)(1− δ)t−1−t′

)B′t′ .

With either of the assumptions m = 1 or δ = 0, m + (1 − m)(1 −δ)t−1−t′ = 1 and the change in the climate state becomes

M ′t =

t−1∑t′=0

B′t′ .

Substituting in the expressions for B′t from (2.77) gives

M ′t =

− 1−βt

1−βt+1

ξX,TξB,T

if t ≤ T

− 1−βt1−βt+1

ξX,TξB,T

+ξX,TξB,T

if T < t ≤ t

0 if t > t

.

The sum that is relevant for the indirect e�ect on welfare in the decen-tralized equilibrium is then

∞∑t=0

βtM ′t =

t∑t=0

βtM ′t = −

t∑t=0

βt − β2t

1− β t+1

ξX,TξB,T

+t∑

t=T+1

βtξX,TξB,T

=−1−βt+1

1−β −1−β2(t+1)

1−β2

1− β t+1

ξX,TξB,T

+1− β t+1 − (1− βT+1)

1− βξX,TξB,T

=−1−βt+1

1−β −(1+βt+1)(1−βt+1)

(1−β)(1+β)

1− β t+1

ξX,TξB,T

+βT+1 − β t+1

1− βξX,TξB,T

=β

1− ββT (1 + β)− (1 + β t+1)

1 + β

ξX,TξB,T

.

2.5. MODEL WITH CAPITAL AND σY = θ = 1 55

The �rst ratio in the last expression is positive. Assuming that βT (1 +β) > 1, the second ratio can be either positive or negative depending ont. It is negative if t = T + 1 and positive if t is large enough.

The conclusion from this analysis is that even in this much simpli-�ed functional-form example there is no simple relationship between thechange in XT and the welfare gains from taxation.

2.5 Model with capital and σY = θ = 1

Above, in propositions 2.4 and 2.6, it was shown that assuming loga-rithmic utility simpli�es the analysis. In this section I will show that ifσY = 1 and capital depreciates fully between periods, then most resultsthat hold for logarithmic utility also hold in a model with capital.5

To begin with, when σY = 1 energy and other inputs are combinedinto �nal goods production according to a Cobb-Douglas productionfunction. It does then not make sense to distinguish between TFP,AY , the productivity of the non-energy inputs (AL or AKL) and theproductivity of the composite energy good, AE: all these terms can bemultiplied together to give a new, all-inclusive productivity factor. Fromtables 2.1 and 2.2 with σY = 1, the sign of the e�ects of changes in thesefactors depend only on the sign of θ − 1. If θ = 1 the e�ects of all thesetechnology factors on fossil-fuel use are zero. This is as before, because ifθ = 1, then the e�ects on marginal utility of consumption and marginalproductivity of fossil fuel exactly balance each other.

The signs of the e�ects of AS, S and B depend on the sign of σE−1.If σE > 1, the e�ects, without externalities, are such that an increasein ASS or a decrease in AB decreases fossil-fuel use in the period wherethe change occurs and increases fossil-fuel use in all other periods (andthe other way around).

So assume now that θ = σY = 1 and that capital depreciates fullybetween periods. Production can then be written

Y = D(M)AY(L1−αKα

)γKL Y γEE ,

where

YE =[γB(ABB)

σ−1σ + γS(ASS)

σ−1σ

] σσ−1

.

De�neα = αγKL

and let

F (K,B;L, S,AY , AB, AS) = AYL(1−α)γKLK αY γE

E ,

5The results that may not apply are those related to the welfare e�ects in section2.4.5 and those with endogenous supply of the alternative-energy input in section2.6.


I will now show that the equilibrium condition for fossil-fuel use inthe decentralized equilibrium and the optimality condition for fossil-fueluse in the planner solution will be the same under these assumptions asin the log utility case of the model without capital (in sections 2.4.2 and2.4.3, respectively).

2.5.1 Decentralized equilibrium with a fossil-fuel tax

The main di�erence here compared to section 2.4.2 is that the house-holds now hold capital. This means that the households receive capitalincome. In addition, the rental rate of capital must also be speci�ed; itwill be denoted rt. Since the model is deterministic and there is full de-preciation, the rental rate of capital will also be the interest rate used forthe discounting of future pro�ts received by fossil-fuel resource owners.

This gives the households' budget constraint

Ct +Kt+1 = wtL+ rtKt + πB,t + πS,t + gt. (2.78)

Apart from these changes, the de�nition of the decentralized equilib-rium is the same as without capital.

A decentralized equilibrium with taxation consists of sequences ofquantities {Bt, Ct, Kt+1}∞t=0, fossil-fuel taxes {τt}∞t=0, lump-sum tax re-bates {gt}∞t=0 and prices {pB,t, pS,t, rt+1, wt}∞t=0 such that

• Households choose their consumption and investments to maxi-mize their discounted utility

∑∞t=0 β

tu(Ct) subject to the budgetconstraint (2.78) for all t.

• Prices are competitively determined.

• Fossil-fuel extracting �rms maximize the discounted pro�t fromextraction.

• The government budget is balanced in each period.

As before, I assume that the alternative-energy input and labor areboth inelastically supplied so that their quantities are not endogenouslydetermined. Their prices will, however, be endogenously determined anda�ect the pro�ts of the suppliers of the alternative-energy input (which inthe end goes to the households) and the labor income of the households.

I will now derive the implications of each of these conditions in turnand then characterize the resulting equilibrium allocation.


Households The utility maximization problem of the representativehousehold is6

max{Ct,Kt+1}∞t=0

∞∑t=0

βtU(Ct) s.t. (2.78).

By substituting for consumption from the budget constraint, the de-cision problem is reduced to choosing next period's capital at all times.Taking the �rst-order condition with respect to Kt+1, one obtains

βtU ′(Ct) = βt+1U ′(Ct+1)rt+1 ⇒1

β

Ct+1

Ct= rt+1. (2.79)

Competitive prices When �nal goods producers act as price takers,the equilibrium prices are

pB,t =D(Mt)FB,t, pS,t = D(Mt)FS,twt =D(Mt)FL,t, rt = D(Mt)FK,t = α Yt

Kt

(2.80)

Firms supplying fossil fuel As in the case without capital, max-imization of discounted pro�ts from fossil-fuel extraction requires thatthe discounted after-tax price of fossil fuel is the same in all periods withpositive fossil-fuel use. The pro�ts are discounted using the rental rateof capital. Since there are no extraction costs for fossil fuel, the pro�tsfrom fossil-fuel extraction is the after tax price of fossil fuel times theextracted quantity. So the situation is exactly the same as in the casewithout capital and, consequently, the pro�t maximization condition isthe same as without capital (2.46). That is,[

t−1∏t′=0

1

rt′+1

](1− τt)pB,t

is the same in all periods with positive aggregate fossil-fuel use, Bt > 0.

Balanced government budget The government balances its budgetin each time period implying that, for each t,

gt = τtpB,t = {(2.80)} = τtD(Mt)FB,t.

6Here only the capital investment decision is included. In principle trade in theshares in energy companies could be included but in equilibrium there will be notrade in these shares. Including these shares would simply allow us to compute theirequilibrium prices.


Equilibrium allocation Since there are no extraction costs for theenergy inputs, the prices pB,t and pS,t directly gives the pro�ts of theenergy supplying �rms

πB,t = (1− τt)pB,tBt = (1− τt)D(Mt)FB,tBt

πS,t = pS,tSt = D(Mt)FS,tSt.

Summing up the income sources for the households

wtlt + rtKt + πB,t + πS,t + Tt =D(Mt)FL,tLt +D(Mt)FK,tKt

+D(Mt)FB,tBt +D(Mt)FS,tSt

=D(Mt)Ft = Yt,

where the last step follows since F has constant returns to scale in L,K, B and S.

Substituting the capital rental rate from (2.80) in the households'�rst-order condition (2.79) delivers

Ct+1

Ct= βα

Yt+1

Kt+1

.

This condition is ful�lled by the consumption/investment rule

Ct = (1− αβ)Yt and Kt+1 = αβYt.

The discount factor for pro�ts in period t ≥ 1 ist−1∏t′=0

1

rt′+1

=t−1∏t′=0

1

α

Kt′+1

Yt′+1

=t−1∏t′=0

1

α

αβYt′

Yt′+1

=t−1∏t′=0

βYt′

Yt′+1

= βtY0

Yt.

Substituting this into the equilibrium condition gives that

βtY0

Yt(1− τt)pB,t = βt

Y0

Yt(1− τt)D(Mt)FB,t

should be the same for all periods with positive fossil-fuel use.Comparing two time periods t1 and t2 such that Bt1 > 0 and Bt2 > 0,

we deduce that

βt1(1− τt1)FB,t1Ft1

= βt2(1− τt2)FB,t2Ft2

. (2.81)

The ratio FB,tFt

is independent of the capital stock. This completes thecharacterization of the competitive equilibrium.

Comparing the condition for fossil-fuel use here to the correspondingexpressions in the model without capital, given by equations (2.5) and(2.48), it can be seen that the expressions are the same. With Cobb-Douglas production, capital cancels from expression (2.81).

This implies that fossil-fuel use will be the same here as in the modelin section 2.4.2 without capital and with logarithmic utility.



The planner solves the problem

max{Bt,Kt+1}

∞∑t=0

βtU(D(Mt)Ft −Kt+1) s.t.∞∑t=0

Bt ≤ Q.

The �rst-order condition with respect to Kt+1 is

βtU ′(Ct) = βt+1U ′(Ct+1)FK,t+1.

With the functions used here this equation delivers

1

Ct= β

1

Ct+1

αYt+1

Kt+1

⇒ Kt+1

Ct= αβ

Yt+1

Ct+1

.

This is ful�lled by having

Ct = (1− αβ)Ft and Kt+1 = αβFt ∀t,

which is the same as in the decentralized equilibrium.The �rst-order condition with respect to Bt, for any t such that

Bt > 0, gives

λ = βtU ′(Ct)D(Mt)FB,t +∞∑

t′=t+1

βt′U ′(Ct′)Ft′D

′(Mt′)dMt′

dBt

,

where λ is the multiplier on the constraint on the total supply of fossilfuel. With the functional forms assumed here, we obtain that for any tsuch that Bt > 0

βtFB,tFt

+∞∑

t′=t+1

βt′D′t′

Dt′

dMt′

dBt

= λ. (2.82)

Comparing conditions (2.82) and (2.56) with θ = 1, we note the samerelative comparison between time periods. As in the decentralized equi-librium, capital cancels out from this expression.

In conclusion, fossil-fuel use will be the same here as in the model insection 2.4.3 without capital and with logarithmic utility.


Comparing the competitive equilibrium with taxation and the plannersolution, the investment choices are the same in both cases in the sensethat the same share of �nal good production is invested in both cases.


This means that a tax system that induces the optimal path of fossil-fueluse in the decentralized equilibrium will induce the social optimum.

Comparing the equilibrium condition in the decentralized equilib-rium with taxation (2.81) to the optimality condition from the plannersolution (2.82), the optimum can be implemented by a tax system thatful�lls

τtFB,tFt

= −∞∑

t′=t+1

βt′−tD

′t′

Dt′

dMt′

dBt

Comparing this to (2.73) with θ = 1, it can be seen that they are equiv-alent. Therefore, the optimal taxes will be the same here as in section2.4.4 without capital and with logarithmic utility.

2.6 Elastic supply of the alternative-energy input

So far, the alternative-energy input, S, has been supplied inelasticallyand using alternative energy has not been associated with any costs.In this section I will consider the case where the provision of S usesresources that could be used elsewhere. To study this I will use a modelwithout capital and without externalities. In my setting, two di�erenttypes of costs are conceivable. One is that extraction of the alternative-energy input requires labor. Another one would be that the use of thealternative-energy input requires some consumption of �nal goods. I willhere focus on the case where the alternative-energy input uses labor.

I assume that one unit of the alternative-energy input uses aS units oflabor. Let the total amount of available labor be exogenously given anddenoted by L. Labor can then either be used to produce the alternative-energy input or to produce �nal goods. Let the amount of labor used inalternative-energy production and �nal-good production be LS and LY ,respectively. These must ful�ll LS + LY = L. The alternative-energyinput is now given by the linear production schedule S = aSL

S. I willassume that labor can move freely between the sectors so that the laborallocation decision can be treated as a static decision made within eachperiod.

Production in period t depends on the amount of fossil fuel used, Bt,the amount of labor used in �nal good production, LYt , the amount ofthe alternative-energy input, St, and on the set of exogenously given vari-ables Γt which now consists of the productivities (AY,t, AL,t, AE,t, AB,t, AS,t),the productivity of labor in producing the alternative-energy input, aS,t,and the total amount of labor L. So, production, which is equal toconsumption, can be written as

Ct = Yt = F (Bt, LYt , St; Γt). (2.83)

2.6. ELASTIC SUPPLY OF ALTERNATIVE ENERGY 61

The production function is the same as in section 2.2 and de�ned inequations (2.11) and (2.12).

Since I am ruling out any externalities, the planner solution andthe decentralized equilibrium will coincide and I will only solve for theplanner solution.


In each period, the planner chooses how much fossil fuel to use, how muchof the alternative-energy input to use and how to allocate labor betweenalternative-energy provision and �nal-goods production. The constraintsare the total supply of fossil fuel, the constraint on total available laborin each period, the production function for the alternative-energy inputand non-negativity constraints on all the chosen variables. This givesthe following planner problem:

max{Bt,St,LYt ,LSt }∞t=0

∞∑t=0

U(F (Bt, L

Yt , St; Γt)

)s.t.

∞∑t=0

Bt ≤ Q0

∀t: LYt + LSt ≤ Lt, St = aS,tLSt

∀t: , LYt ≥ 0, LSt ≥ 0, Bt ≥ 0.


L=∞∑t=0

βtU(F (Bt, L

Yt , St; Γt)

)+ λ

[Q0 −

∞∑t=0

Bt

]

+∞∑t=0

µL,t[Lt − LYt − LSt

]+∞∑t=0

µS,t[aS,tL

St − St

]+∞∑t=0

[ηY,tL

Yt + ηS,tL

St + ηB,tBt

]Taking �rst-order conditions, we obtain

Bt : βtU ′(Ct)FB,t = λ− ηB,tSt : βtU ′(Ct)FS,t = µS,t

LYt : βtU ′(Ct)FL,t = µL,t − ηL,tLSt :µL,t = µS,taS,t + ηS,t.

I will now assume that σE <∞, that is, that the energy sources arenot perfect substitutes. This will imply that the production function


ful�lls Inada conditions for B, S and LY so that non-negativity con-straints will never bind. That is ηY,t = ηS,t = ηB,t = 0. Furthermore, themarginal product of all inputs will always be strictly positive implyingthat the inequality constraints will hold with equality. The �rst-orderconditions with respect to St, LYt and LSt can now be combined to give

FL,t = aS,tFS,t, (2.84)

which means that the marginal product of labor must be the same inboth possible uses in all periods.

Since Bt > 0 for all t, the �rst order condition with respect to Bt

implies that, for any t1 and t2,

βt1U ′(Ft1)FB,t1 = βt2U ′(Ft2)FB,t2 ,

which, as before, says that the marginal value of fossil-fuel use shouldbe the same in all periods. For a given t, U ′(Ct)FB,t will depend on thesame variables as appear in the production function, that is, Bt, LYt ,St = aS,tL

St and Γt. Along the same lines as before it is now possible to

de�neH(Bt, L

Yt , aS,tL

St ; Γt) = U ′(Ft)FB,t.

However, since (2.84) determines the intratemporal allocation of laborbetween the sectors, and implicitly de�nes LYt and LSt as functions ofBt and Γt, H can be written as a function of only Bt and Γt. So I willde�ne

H(Bt; Γt) =H(Bt, LYt , aS,tL

St ; Γt)

where LYt + LSt = Lt and (2.84) is ful�lled. (2.85)

Using this de�nition, the equilibrium condition, for any t1 and t2 is

βt1Ht1 = βt2Ht2 (2.86)

2.6.2 Changes in {Γt}∞t=0

The equilibrium condition is now very similar to (2.6) and the e�ects ofvarying the exogenous variables in ΓT can be analyzed in much the sameway. The main di�erence is that when di�erentiating H with respect toeither of its arguments, the e�ect on the labor allocation must also betaken into account. I will now calculate the e�ect on Ht of changing Bt.

Firstly, to �nd the e�ect on the labor allocation decision, I will dif-ferentiate condition (2.84) with respect to Bt and treat LSt and LYt asfunctions of Bt (and Γt). Since all variables are in the same time period,


I will suppress the time indices. Di�erentiating both sides of (2.84) withrespect to B gives

aS[FSB + FSSaSL

SB − FSLLSB

]= FLB + FLSaSL

SB − FLLLSB,

where I have used that LYB = −LSB. Solving for LSB gives

LSB =aSFSB − FLB

2aSFLS − FLL − a2SFSS

. (2.87)

When di�erentiating H with respect to Bt, all time indices are the sameand I will suppress them. The derivative with respect to B is then

∂H

∂B=U ′′(F )FB

[FB + FSaSL

SB − FLLSB

]+U ′(F )

[FBB + FBSaSL

SB − FBLLSB

]=U ′(C)

[FBB − θ

F 2B

F+ (FBSaS − FBL)LSB

].

The optimality condition (2.84) was used to cancel terms from the �rstparenthesis and the form of the utility function (2.10) was used to sub-stitute for U ′′(C).

Substituting LSB from (2.87) gives that

∂H

∂B= U ′(C)

[FBB − θ

F 2B

F+

(FBSaS − FBL)2


]. (2.88)

It can be shown that this derivative is negative (see equation (2.102)in appendix 2.A.4). This leads to the following proposition.

Proposition 2.9. If σY > 0 and σE < ∞, then Bt > 0 for all t, thelabor allocation is interior for all t and the e�ects on fossil-fuel use of achange in XT is

Sgn

(dBT

dXT

)= Sgn

(∂HT

∂XT

)and

Sgn

(dBt

dXT

)= Sgn

(−∂HT

∂XT

)for all t 6= T

Proof. When σY > 0 and σE <∞, the production function ful�lls Inadaconditions for all its inputs and there will be an interior solution in allperiods. Starting from the equilibrium condition (2.86) and the obser-vation that ∂Ht

∂Bt< 0 for all t, the rest of the proposition follows from the

same logic as the proof of proposition 2.1.


In conclusion, the e�ect of a change in the technology factor XT

depends on the partial e�ect of the change on HT . Since all variablesconsidered in what follows, will concern period T , I will suppress thetime indices from the notation.

To begin with, the e�ect of the change in X on the labor allocationmust be found. Since aS appears explicitly in expression (2.84), di�er-entiating with respect to as is qualitatively di�erent than di�erentiatingwith respect to any of the other technology factors. I will start by look-ing at changes in X ∈ {AY , AL, AE, AB, AS}. Taking the dependency ofLS and LY on X into account and di�erentiating both sides of the laborallocation condition (2.84) with respect to X gives

aS[FSX + FSSaSL

SX − FSLLSX

]= FLX + FLSaSL

SX − FLLLSX ,

where I have used that LYX = −LSX . Solving for LSX gives

LSX =aSFSX − FLX


. (2.89)

Di�erentiating H with respect to X then yields

∂H

∂X=U ′′(F )FB

[FX + FSaSL

SX − FLLSX

]+U ′(F )

[FBX + FBSaSL

SX − FBLLSX

]=U ′(C)

[FBX − θ

FBFXF

+ (FBSaS − FBL)LSX

]. (2.90)

Here, the optimality condition (2.84) was used to cancel terms from the�rst parenthesis and the form of the utility function (2.10) was used tosubstitute for U ′′(C).

Looking at the expression within the parenthesis, it consists of twoparts. The �rst part, consisting of two terms, is the same as the expres-sion determining the sign in the case with inelastic supply of the cleanenergy input (2.17). The second part relates to the e�ect of the changein the allocation of labor. An increase in LS increases the amount of thealternative-energy input and decreases the amount of labor used in �nalgoods production. The derivatives are (see appendix 2.A.4)7

FBS =FB

[1

σE− 1

σY

GL

GE +GL

]GS

GB +GS

1

S

FBL =FB1

σY

GL

GL +GE

1

LY.

7In all the following calculations, the Gs are de�ned as in (2.16).


The �rst derivative is ambiguous since it depends on the size of GLGE+GL

The positive part of this comes from the fact that an increase in theamount of alternative energy increases the productivity of fossil fuelin producing the composite energy good, while the negative part comesfrom the fact that there is more energy in relation to the complementaryinput labor, which decreases the marginal product of energy. The secondderivative is positive indicating that an increase in the amount of laborused in �nal goods production increases the marginal product of fossilfuel. The combination of the derivatives is (see appendix 2.A.4)

aSFBS − FBL = FB

(1

σE− 1

σY

)GS

GB +GS

aSS.

This expression is negative under assumption (2.13). That is, the totale�ect of moving labor from �nal good production to production of thealternative-energy input is to decrease the marginal product of fossil fuel.

From the calculations in the appendix (see 2.A.4) it follows thatLSAY = 0, LSAL ≥ 0, LSAE ≤ 0 and LSAB < 0 while the sign of LSAS isambiguous. As long as FBX − θFBFX

Fand LSX has opposite signs, the

sign of ∂H∂X

is unambiguously determined. Combining table 2.1 with thesigns of LSX , it follows that, under assumptions (2.13)-(2.15)

∂H

∂AY≤ 0.

For AB, the sign of FBAB − θFBFAB

Fis ambiguous and for AS, the sign

of LSAS is ambiguous. So for all these factors the partial derivative of Hmust be calculated.

Substituting LSX from (2.89) into (2.90) gives

∂H

∂X= U ′(C)

[FBX − θ

FBFXF

+(aSFBS − FBL) (aSFSX − FLX)


].

(2.91)The calculation of these derivatives can be found in the appendix (see2.A.4). The derivatives that are unambiguously determined by assump-tions (2.13)-(2.15) are

∂H

∂AY≤ 0,

∂H

∂AE< 0 and

∂H

∂AS< 0 (2.92)

Considering a change in AL, the derivative of H with respect to AL is


(see appendix 2.A.4 for more detail)

∂H

∂AL=u′(C)

FBAL

1σY

(1 + 1−σY

σE− θ)

GSGB+GS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

aSS

+u′(C)FBAL

(1σY− θ)

1σE

GLGE+GL

GBGB+GS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

aSS

The sign of this expression is, in general, ambiguous, re�ecting the twoopposing e�ects. If θ ≤ 1 + 1−σY

σE> 1 it is unambiguously positive. If

θ > 1+ 1−σYσE

, the �rst term is negative while the second term is positive.

Considering a change in AB, the derivative of H with respect to ASbecomes (see appendix 2.A.4 for more detail)

1

u′(C)

∂H

∂AB=FBAB

σE − 1

σE

GS

GB +GS

−FBAB

(1− σYσY

GL

GE +GL

+ (θ − 1)GE

GE +GL

)GB

GB +GS

+FBAB

(1σE− 1

σY

)2GB

GB+GS

GSGB+GS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

aSS.

The �rst two terms are the same as in the case with inelastic supply ofS. The third term is positive. If GB << GS or GB >> GS the thirdterm is small and the results that the derivative is positive if GS >> GB

and negative if GB << GS holds also with elastic supply of S.I will now turn to the e�ects of a change in aS. Di�erentiating con-

dition (2.84) with respect to aS gives

FLS(aSL

SaS

+ LS)− FLLLSaS =FS + aS

(aSFSSL

SaS− FSLLSaS + LSFSS

),

where I have used that LYaS = −LSaS . Solving for LSaS gives

LSaS =FS + aSL

SFSS − LSFLS2aSFLS − a2

SFSS − FLL. (2.93)

A change in aS a�ects the marginal value of fossil-fuel use both by af-fecting the amount of alternative energy available, for a given amountof labor used in the production of the alternative-energy input, and bya�ecting the allocation of labor. Di�erentiating H with respect to aS

2.7. DISCUSSION 67

gives

∂H

∂aS=U ′′(F )

[LSFS − LSaSFL + aSFSL

SaS

]FB

+U ′(F )[LSFBS − LSaSFBL + aSFBSL

SaS

]=U ′(F )

[(FBS − θ

FBFSF

)LS + (aSFBS − FBL)LSaS

],

where the labor allocation condition (2.84) and the shape of the utilityfunction (2.10) were used. Substituting for LSaS gives

∂H

∂aS=U ′(F )

(FBS − θ

FBFSF

)LS

+U ′(F )

(FS + aSL

SFSS − LSFLS)

(aSFBS − FBL)

2aSFLS − a2SFSS − FLL

. (2.94)

As shown in appendix 2.A.4, this expression is negative under assump-tions (2.13)-(2.15).

This can all be summarized in the following proposition:

Proposition 2.10. Under the assumptions of proposition 2.9 and as-sumptions (2.13)-(2.15) and for t 6= T ,

dBT

dXT

≤ 0 anddBt

dXT

≥ 0

for X ∈ {AY , AE, AS, aS} while the e�ect of varying AL or AB is am-biguous.

Thus, compared to the case with inelastic supply of the alternative-energy input, the e�ects are the same except for the fact that the e�ectof the labor augmenting technology AL is now ambiguous. Furthermore,the e�ects of varying as on fossil-fuel use have the same sign as the e�ectsof varying AS.

2.7 Discussion

In this chapter I have investigated how the trends of technological devel-opment a�ects the intertemporal pattern of fossil-fuel use. I have alsoconsidered the robustness of the results to a number of variations of theassumptions. Apart from the results regarding the technology that iscomplementary to energy, in the case where the complementary inputcould also be used to produce alternative energy, the qualitative resultshave remained the same throughout the analysis. In this section, I will


discuss four di�erent aspects of the assumptions I have made. I will �rstbrie�y discuss three assumptions that I have made and that I do notthink a�ect the results much. I will then discuss the assumption aboutzero extraction costs of fossil fuel in somewhat more detail. In particu-lar, I will show, using a simple example, that this assumption does seemto be important.

Throughout, I have treated technological developments as exogenous.In reality, they are driven by forward-looking decisions. Endogenizingtechnology could potentially provide interesting insights regarding theinteraction between the fossil-fuel supply and forward-looking researchactivity. In such a model, subsidies to research on the di�erent types oftechnology could be studied instead of exogenous changes in the tech-nology factors. However, while the interaction between the di�erentdecisions made could amplify or damped the mechanisms studied here,it does not seem likely that the signs of the e�ects should change.

Except for the anlyses in sections 2.3 and 2.5, the treatment hereabstracts from capital accumulation. While the omission simpli�es theanalysis, I do not think that it a�ects the qualitative results much. Thetwo sections where I do include capital (but restrict the models in otherways) con�rm the results from the other sections. In section 2.5 I alsoprovide results for an in�nite time horizon model with θ = σY = 1. Morerealistic assumptions would probably involve a θ that is a little bit largerthan one, while σY would be lot smaller. From both tables 2.1 and 2.2it can be seen that changes away from the assumptions of section 2.5toward more realistic values should strengthen the e�ects I derive. Vander Ploeg and Withagen (2011) also �nd a Weak Green Paradox in agrowth model with capital.

In section 2.4 I consider di�erent speci�cations of the descriptionof the carbon cycle. There I derive results under somewhat restrictiveassumptions. While it may be di�cult to generalize the analysis analyti-cally, assumptions, the proofs rely on a sequence of su�cient conditions.This suggests that the results may well apply under much more generalassumptions. In particular, for most of the results, I assumed that eitherδ = 0 or that m = 1. The implications of these assumptions that aremost important for the results are that they imply that the climate stateis increasing over time and that changes in emissions accumulate overtime. In reality, the climate state will keep increasing for a long time.So that part seems unproblematic. It also seems reasonable that, if com-paring two di�erent emission paths where one has higher emissions fora couple of decades, the di�erence during this time between the inducedclimate paths should also increase over time.

A potentially more critical assumption throughout my analysis is

2.7. DISCUSSION 69

that fossil fuel is costlessly extracted from a given total supply. Ex-traction decisions are then based completely on the relative pro�tability(or marginal value in planner solutions) between extraction at di�erentpoints in time. This means that the scarcity value of the resource isvery important. This may not be too bad an assumption for oil, but thesupply decisions for coal are more likely to be determined by extractioncosts than by resource scarcity. My assumption was made to emphasizethe mechanisms studied here. That is, the way I model fossil-fuel supplyhighlights the way that scarcity rents in�uence the results. It is, how-ever, not an unproblematic assumption. See Hart and Spiro (2011) fora critical discussion of the role of the scarcity rent in the price of oil andcoal. Having extraction costs for fossil fuel that increase as the remain-ing resources decrease has been studied in the Green Paradox literature(see, e.g., van der Werf and Di Maria (2011)). The typical result there isthat a decrease in the future value of fossil-fuel use results in more fossil-fuel use in the short run but less total fossil-fuel use. In the terminologyof Gerlagh (2011), this means that there is still a Weak Green Paradoxbut that the existence of a Strong Green Paradox is less obvious.

Smulders et al. (2010) �nd that a green paradox can arise also withoutscarcity. They, however, consider the e�ects of announcing a futuretax on fossil-fuel use rather than an improvement in the future state ofalternative-energy technology. The results there are driven by the factthat the reduction in future production, caused by taxation, increasesthe value of investments, which in turn induces more fossil-fuel use inthe short run to make investments. When considering an improvementin the future state of technology, this investment e�ect should go in theother direction. I will now demonstrate this using a simple example.

The assumption of zero extraction costs is on one of the extreme endsof the spectrum of possible assumptions in the sense that only scarcity,and not extraction costs, matters. I will now go to the other extreme andassume that scarcity plays no role at all and that extraction of fossil fuelis associated with extraction costs that are independent of the remainingstock. I will keep the model as simple as possible. In order to capturethe involved dynamics, I need to have capital in the model and I need(at least) three time periods. I will not include labor. I will assume thatcapital can be used for three di�erent activities: directly in �nal goodsproduction, in extraction of fossil fuel and in generation of alternativeenergy. I will assume that both fossil-fuel extraction and alternative-energy generation are linear in the amount of capital used. The onlytechnology factor that I will consider is the technology for alternative-energy generation. I will assume that capital can be reallocated freelywithin a period so that the allocation decision is static. Let the total


amount of capital, in a period, beK and the amount used in the di�erentsectors be KY , KB and KS, respectively.

Production in a period is given by

Y = FY (KY , E) =

[K

σY −1

σYY + E

σY −1

σY

] σYσY −1

,

where

E = FE(B, S) =[B

σE−1

σE + SσE−1

σE

] σEσE−1

andB = KB, S = ASKS.

The division of K into KY , KB and KS can be shown to be

KY =

[1 + AσE−1

S

]σE−σYσE−1

1 + AσE−1S +

[1 + AσE−1

S

]σE−σYσE−1

K

KB =1

1 + AσE−1S +

[1 + AσE−1

S

]σE−σYσE−1

K

KS =AσE−1S

1 + AσE−1S +

[1 + AσE−1

S

]σE−σYσE−1

K.

The important observation here is that

∂B

∂K=∂KB

∂K> 0 and

∂B

∂AS=∂KB

∂AS< 0. (2.95)

That is, the more capital there is, the more capital is used for fossil-fuelextraction and therefore the higher is fossil-fuel use. Also, the better isthe technology for alternative-energy generation, the less capital is usedfor fossil-fuel extraction and, therefore, the lower will fossil-fuel use be.

Final good production can then be shown to be

Y =

[1 +

(1 + AσE−1

S

)σY −1

σE−1

] 1σY −1

K ≡ F (AS)K,

where

F (AS) =

[1 +

(1 + AσE−1

S

)σY −1

σE−1

] 1σY −1

and F ′(AS) > 0.

I will only be interested in varying AS,3. I will therefore set F (AS,1) =

F (AS,2) = F .

2.8. CONCLUDING REMARKS 71

Assuming full depreciation of capital, the intertemporal optimizationproblem is now

maxK2,K3

U(FK1 −K2

)+ βU

(FK2 −K3

)+ β2U

(F (AS,3)K3

).

The �rst order conditions are

K2 :U ′(C1) = βFU ′(C2)

K3 :U ′(C2) = βF (AS,3)U ′(C3) = βF (AS,3)1−θK−θ3 .

Consider an increase in AS,3: it will increase the marginal productof capital and decrease the marginal utility of consumption. Assumingthat θ > 1, the net e�ect is a decrease in the marginal value of capitalin period 3. In order to maintain the �rst-order conditions at equality,consumption has to increase in periods 1 and 2. Increasing consump-tion in period 1 means decreasing investment, implying that K2 mustdecrease. If second-period consumption increases, while second-periodcapital decreases, then second-period investment also must decrease.

Thus, the conclusion is that if AS,3 increases, we will observe de-creases in K2 and K3. Using the derivatives (2.95), this implies that�rst-period fossil-fuel use is unchanged while second and third-periodfossil-fuel use decreases. That is, the results of the green paradox donot hold here. An increase in the future state of alternative-energy tech-nology leads to a decrease in fossil-fuel use in both the short and thelong run. Since the results here rely partly on the value of investments,there should be a similar e�ect for an improvement in any technologyfactor. There will also be an e�ect of intratemporal reallocation of capi-tal within the period. If the technology for fossil-fuel extraction changes,there would also be a direct e�ect.

2.8 Concluding remarks

In this chapter I have analyzed how technological trends a�ect the in-tertemporal pattern of fossil-fuel use. Throughout the chapter I haveassumed that fossil fuel was costlessly extracted from a given total sup-ply. Under that assumption, the conclusions seem to be robust.

Regarding the Green Paradox, the results derived here con�rms theexistence of a Weak Green Paradox, at least as long as the fossil-fuelsupply is driven by scarcity. A future improvement in either technologyfor alternative-energy generation or energy-saving technology leads toincreased fossil-fuel use in the short run.

My study also emphasizes that the developments of other technol-ogy factors, as the result of increased spending on a particular type of


technology, also matter for the ultimate e�ects on fossil-fuel use. Thatis, if increased research into one type of technology crowds out researchof other types of technologies, this crowding out must also be taken intoaccount. Except for research on technology that is complementary toenergy (and possibly the technology for using fossil fuel), this crowdingout dampens the e�ects of the Green Paradox.

However, the example in the discussion indicates that the results canbe changed signi�cantly, and possibly reversed if the supply of fossil fuelis driven by extraction costs rather than by the scarcity value.

References

Archer, D., 2005, �The Fate of Fossil Fuel CO2 in Geologic Time�, Jour-nal of Geophysical Research, 110(C9).

Dasgupta, P. & G. Heal, 1974, �The Optimal Depletion of ExhaustableResources�, Review of Economic Studies, 41, 3-28.

European Commission, 2011, �A roadmap for moving to a competitivelow carbon economy in 2050�. Available at:http://eur-lex.europa.eu/LexUriServ/LexUriServ.do?uri=COM:2011:0112:FIN:EN:PDF.

Gerlagh, R., 2011, �Too much oil�, CESifo Economic Studies vol 57,1:79-102.


Hart, R. & D. Spiro, 2011, �The elephant in Hotelling's room�, EnergyPolicy, 39, 7834-7838.

Hassler, J., P. Krusell & C. Olovsson, 2011, �Energy-Saving TechnicalChange�, mimeo , Institute for International Economic Studies, Stock-holm University.

Hotelling, H., 1931, �The Economics of Exhaustible Resources�, TheJournal of Political Economy, 39:2, 137-175.

Layard, R., S. Nickell & G. Mayraz, 2008, �The marginal utility of in-come� Journal of Public Economics 92:1846-1857.

Sinclair, P., 1992, �High does nothing and rising is worse: carbon taxesshould keep declining to cut harmful emissions� The Manchester School,LX(1), 41-52.

Sinn, H.-W., 2008, "Public policies against global warming: a supply sideapproach", International Tax and Public Finance, Volume 15:4, 360-394.

Smulders, S., A., Y. Tsur, & A. Zemel, 2010, �Announcing Climate Pol-icy: Can a Green Paradox Arise Without Scarcity?� CESifo WorkingPaper Series No. 3307.Available at SSRN: http://ssrn.com/abstract=1742789.

Van der Ploeg, F. & C. Withagen, 2011, �Growth, Renewables and the

73


Optimal Carbon Tax�, OxCarre Research Paper 55.

Van der Ploeg, F. & C. Withagen, 2012, �Is there really a green para-dox?�, Forthcoming in Journal of Environmental Economics and Man-agement.

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2.A. CALCULATIONS 75

2.A Calculations

2.A.1 Calculations for model without capital

In order to calculate the expression (2.17) for the exogenous variables,start by de�ning Y as

Y = AY Y ,

where Y is given by (2.11).Using (2.11) and (2.12) the following derivatives can be calculated:

∂Y

∂(AEYE)= Y

1σY γE(AEYE)

− 1σY = Y

1σY

GE

AEYE(2.96)

∂Y

∂(ALL)= Y

1σY γL(ALL)

− 1σY = Y

1σY

GL

ALL(2.97)

∂YE∂(ABB)

=Y1σEE γB(ABB)

− 1σE = Y

1σEE

GB

ABB(2.98)

∂YE∂(ASS)

=Y1σEE γS(ASS)

− 1σE = Y

1σEE

GS

ASS, (2.99)

where the Gs are de�ned in (2.16). Note that, for instance,

∂Y

∂AE=

∂Y

∂(AEYE)

∂AEYE∂AE

= Y1σY γE(AEYE)

− 1σY YE = Y

1σYGE

AE.

Using these derivatives, the marginal product of fossil fuel is

FB =AY∂Y

∂B= AY

∂Y

∂YE

∂YE∂B

=AY Y1σY γEA

σY −1

σYE Y

1σE− 1σY

E γBAσE−1

σEB B

− 1σE .

The sign of expression (2.17) is the same as the sign of FBX−θFBFXY.

Thus, in order to compute this expression I need the derivatives of Y = Fand FB with respect to AY , AL, AE, AB, AS and S. I will calculate eachof these in turn.

Derivatives with respect to X = AY :

FAY = Y

FBAY =FBAY

giving

FBAY − θFBFAYY

=FBAY

(1− θ).


Derivatives with respect to X = AL:

FAL =AY∂Y

∂AL

FBAL =FB1

σY

1

Y

∂Y

∂AL

giving

FBAL − θFBFALY

=FB1

σY

1

Y

∂Y

∂AL− θFBAY

Y

∂Y

∂AL

=FB

Y

(1

σY− θ)∂Y

∂AL.

Derivatives with respect to X = AE:

FAE =AY∂Y

∂AE= AY Y

1σYGE

AE

FBAE =FB

[1

σY

1

Y

∂Y

∂AE+σY − 1

σY

1

AE

]

=FB

[1

σYAY Y

1−σYσY

GE

AE+σYσY

1

AE

]=FBY

1−σYσY

1

AE

[σY − 1

σYGL +GE

]giving

FBAE − θFBFAEY

=FBY1−σYσY

1

AE

[σY − 1

σYGL +GE

]− θFBAY Y

1σY

Y

GE

AE

=FBY

1−σYσY

AE

[σY − 1

σYGL + (1− θ)GE

].

Derivatives with respect to X = AB:

FAB =AY∂Y

∂AB= AY

∂Y

∂YE

∂YE∂AB

= AY Y1σY

GE

GL +GE

GB

AB

FBAB =FB

[1

σY

1

Y

∂Y

∂AB+

(1

σE− 1

σY

)1

YE

∂YE∂AB

+σE − 1

σE

1

AB

]


giving

FBAB − θFBFABY

=

= FB

[1

σY

1

Y

∂Y

∂AB+

(1

σE− 1

σY

)1

YE

∂YE∂AB

+σE − 1

σE

1

AB

]− θ

FBAY∂Y∂AB

Y

= FB

[(1

σY− θ)

1

Y

∂Y

∂AB+

(1

σE− 1

σY

)1

YE

∂YE∂AB

+σE − 1

σE

1

AB

]

= FB

[((1

σY− θ)

GE

GL +GE

+

(1

σE− 1

σY

))1

YE

∂YE∂AB

+σE − 1

σE

1

AB

]

=FBAB

(

1σE− 1

σY

)GL +

(1σE− θ)GE

GL +GE

GB

GB +GS

+σE − 1

σE

GB +GS

GB +GS

=FBAB

[σY −1σY

GL + (1− θ)GE

]GB + σE−1

σE(GL +GE)GS

(GL +GE) (GB +GS).

Derivatives with respect to X = AS or X = S:

FX =AY∂Y

∂X= AY

∂Y

∂YE

∂YE∂X

FBX =FB

[1

σY

1

Y

∂Y

∂X+

(1

σE− 1

σY

)1

YE

∂YE∂X

]giving

FBX − θFBFXY

=FB

[1

σY

1

Y

∂Y

∂X+

(1

σE− 1

σY

)1

YE

∂YE∂X

]− θFB

YAY

∂Y

∂X

=FB

[(1

σY− θ)

1

Y

∂Y

∂X+

(1

σE− 1

σY

)1

YE

∂YE∂X

]

=FB

[(1

σY− θ)

1

Y

∂Y

∂YE+

(1

σE− 1

σE

)1

YE

]∂YE∂X

=FBYE

(1σE− 1

σY

)GL +

(1σE− θ)GE

GL +GE

∂YE∂X

.

2.A.2 Calculations for model with capital

I will now calculate (2.23) and (2.24) with X equal to AY , AKL, AE, AB,AS and S. These expressions contain the partial derivatives FB, FK ,


FBK , FKK FX , FBX and FKX . I will start by calculating the derivativeswith respect to the endogenous variables. After that, I will calculatethe derivatives involving each of the exogenous variables in turn. Foreach X, I then combine the derivatives into the expressions in (2.23)and (2.24).

Production is

Y = F (B,K; Γ) = AY Y ,

where

Y =[γKL (AKLYKL)

σY −1


σY

] σYσY −1

,

with

YKL = KαL1−α and YE =[γB (ABB)

σE−1

σE + γS (ASS)σE−1

σE

] σEσE−1

.

Using the de�nitions of the Gs in (2.16) and (2.25), the followingderivatives can be calculated:

∂Y

∂(AFYF )= Y

1σY

GF

AFYFfor F ∈ {E,KL}

∂YE∂(AFF )

= Y1σEE

GF

AFFfor F ∈ {B, S}

and

∂YKL∂K

= αYKLK


Derivatives with respect to endogenous variables

FB =AY∂Y

∂YE

∂YE∂B

= AY Y1σY γEA

σY −1

σYE Y

1σE− 1σY

E γBAσE−1

σEB B

− 1σE

=YGE

GKL +GE

GB

GB +GS

1

B

FBB =FB

(1σY

Y

∂Y

∂B+

1σE− 1

σY

YE

∂YE∂B−

1σE

B

)

=−FB[

1

σY

GKL

GE +GKL

GB

GB +GS

+1

σE

GS

GB +GS

]1

B

FK =AY∂Y

∂YKL

∂YKL∂K

= AY Y1σY γ

σY −1

σYKL Y

σY −1

σYKL α

1

K= Y

GKL

GKL +GE

α1

K

FKK =FK

(1σY

Y

∂Y

∂YKL

∂YKL∂K

+

σY −1σY

YKL

∂YKL∂K

− 1

K

)

=−FK[1− α + α

1

σY

GE

GE +GKL

]1

K

FBK =FB1

σY

1

Y

∂Y

∂K= FK

1σY

Y

∂dY

∂B=

1

σY

FBFKY

Calculations of (2.23) and (2.24) for each X

Calculations for X = AY :

FAY = Y

FBAY =FBAY

=∂Y

∂YE

∂YE∂B

FKAY =FKAY

=∂Y

∂YKL

∂YKL∂K

This deliversFKAYFK

− FBAYFB

=1

AY− 1

AY= 0

and(FKKFK

Y − θFK)FBAYFB

−(

1

σY− θ)FKAY + θ

(FBKFB− FKK

FK

)FAY

=1

AY

[FKKFK

Y − θFK −(

1

σY− θ)FK + θ

(FBKFB− FKK

FK

)Y

]=θ − 1

AY

[1

σYFK −

FKKFK

Y

].

The parenthesis in the last expression is positive, so the expression hasthe same sign as θ − 1.


Calculations for X = AKL:

FAKL =AY∂Y

∂AKL= AY Y

1σYGKL

AKL

FBAKL =FB

σY Y

∂Y

∂AKL=FBσY

Y1−σYσY

GKL

AKL

FKAKL =FK

(1

σY Y

∂Y

∂AKL+σY − 1

σY

1

AKL

)

=FKAKL

Y1−σYσY

(GKL +

σY − 1

σYGE

)This yields

FBAKL −FBFK

FKAKL =FBσY

Y1−σYσY

GKL

AKL

−FBFK

FK Y1−σYσY

AKL

(GKL +

σY − 1

σYGE

)=

1− σYσY

FBAKL

and(FKKFK

Y − θFK)FBAKLFB

−(

1

σY− θ)FKAKL + θ

(FBKFB− FKK

FK

)FAKL

=

(1

σY− θ)Y

1−σYσY

AKL

[FKKFK

Y GKL − FKGKL − FKσY − 1

σYGE

]=

(θ − 1

σY

)Y

1σY

AKL

GKL

KAY .

Calculations for X = AE:

FAE =AY∂Y

∂AE= AY Y

1σYGE

AE

FBAE =FB

(1

σY

1

Y

∂Y

∂AE+σY − 1

σY

1

AE

)

=FBAE

Y1−σYσY

(σY − 1

σYGKL +GE

)FKAE =FK

1

σY

1

Y

∂Y

∂AE=FKσY

Y1−σYσY

GE

AE


This leads to

FBAE −FBFK

FKAE =FBAE

Y1−σYσY

(σY − 1

σYGKL +GE

)− FBσY

Y1−σYσY

GE

AE

=σY − 1

σY

FBAE

and(FKKFK

Y − θFK)FBAEFB

−(

1

σY− θ)FKAE + θ

(FBKFB− FKK

FK

)FAE

=Y

1−σYσY

AE

FKKFK

Y(σY −1σY

GKL + (1− θ)GE

)−θ σY −1

σYFK Y

σY −1

σY −(

1σY

(1σY− θ))

FKGE

=Y Y

1−σYσY

AEK

[1− σYσY

(1 + α (θ − 1))GKL + (θ − 1)

(1 + α

1− σYσY

)GE

].

Calculations for X = AB:

FAB =AY∂Y

∂AB= AY Y

1σY GEY

1−σEσE

E

GB

AB

FBAB =FB

[1

σY

1

Y

∂Y

∂AB+

(1

σE− 1

σY

)1

YE

∂YE∂AB

+σE − 1

σE

1

AB

]

=FB

[− 1

σY

GKL

GE +GKL

GB

GB +GS

+

(1− 1

σE

GS

GB +GS

)]1

AB

FKAB =FK1

σY

1

Y

∂Y

∂AB= FK

1

σY

Y1−σYσY Y

1−σEσE

E

ASGEGB

This delivers

FBAB −FBFK

FKAB =FB

[1

σY

1

Y

∂Y

∂AB+

(1

σE− 1

σY

)1

YE

∂YE∂AB

]

+FBσE − 1

σE

1

AB− FBFK

FK1

σY

1

Y

∂Y

∂AB

=FBY

1−σEσE

E

AB

[σY − 1

σYGB +

σE − 1

σEGS

]


and

(FKKFK

Y − θFK)FBABFB

−(

1

σY− θ)FKAB + θ

(FBKFB− FKK

FK

)FAB

=FKKFK

Y

( 1

σY− θ)

1

Y

∂Y

∂AB+Y

1−σEσE

E

AB

(σY − 1

σYGB +

σE − 1

σEGS

)−FK

( 1

σY− θ)

1

σY

1

Y

∂Y

∂AB+ θ

Y1−σEσE

E

AB

(σY − 1

σYGB +

σE − 1

σEGS

)

=−

(

(θ − 1)(

1 + α 1−σYσY

)GE

GKL+GE

+σY −1σY

(1 + α(θ − 1)) GKLGKL+GE

)GB

GB+GS

+

((1 + α(θ − 1)) GKL

GKL+GE

+(

1 + α 1−σYσY

)GE

GKL+GE

)σE−1σE

GSGB+GS

YK 1

AB.

Calculations for X ∈ {AS, S}:

FX =AY∂Y

∂X= AY Y

1σY GEY

1−σEσE

E

GS

X

FBX =FB

[1

σY

1

Y

∂Y

∂YE+

(1

σE− 1

σY

)1

YE

]∂YE∂X

=FB

[1

σE− 1

σY

GKL

GKL +GE

]GS

GB +GS

1

X

FKX =FK1

σY

1

Y

∂Y

∂X= FK

1

σY

GE

GKL +GE

GS

GB +GS

1

X

This gives

FBX −FBFK

FKX =FB

[1

σE− 1

σY

GKL

GKL +GE

]GS

GB +GS

1

X

−FBFK

FK1

σY

GE

GKL +GE

GS

GB +GS

1

X

=

(1

σE− 1

σY

)FBX

GS

GB +GS


and

(FKKFK

Y − θFK)FBXFB−(

1

σY− θ)FKX + θ

(FBKFB− FKK

FK

)FX

=

FKKFK

Y(

1σE− 1

σY

GKLGKL+GE

− θ GEGKL+GE

)+(θ(

1σY− 1

σE

)+ 1

σY

(θ − 1

σY

)GE

GKL+GE

)FK

GS

GB +GS

1

X

=

(θ − 1

σE

)(1 + α 1−σY

σY

)GE

GKL+GE

+(

1σY− 1

σE

)(1 + α (θ − 1)) GKL

GKL+GE

GS

GB +GS

Y

K

1

X.

2.A.3 Derivation of equation (2.71)

Substituting (2.70) into (2.69) gives

βt2H1,t2

[(1− θ)

D′t2Dt2

(M ′

t1+B′t1

)− ξB,t2B′t2

]−βt1H1,t1

[(1− θ)

D′t1Dt1


]=H2,t1,t2(1− θ)FB,t2

Ft2B′t2

+

t2∑t=t1+1

H2,t1,t


+ (1− θ)D′t

Dt

)(M ′

t1+B′t1

).

Solving for B′2 then leads to

B′t2 =βt2H1,t2(1− θ)D

′t2

Dt2+ βt1H1,t1ξB,t1

βt2H1,t2 +H2,t1,t2(1− θ)FB,t2Ft2

B′t1

−

∑t2t=t1+1 H2,t1,t


Dt+ (1− θ)D

′t

Dt

)βt2H1,t2 +H2,t1,t2(1− θ)FB,t2

Ft2

B′t1

+βt2H1,t2(1− θ)D

′t2

Dt2− βt1H1,t1(1− θ)D

′t1

Dt1


M ′t1

−

∑t2t=t1+1 H2,t1,t


Dt+ (1− θ)D

′t

Dt

)βt2H1,t2 +H2,t1,t2(1− θ)FB,t2

Ft2

M ′t1.


Adding and subtracting βt2H1,t2(1− θ)D′t1

Dt1and

∑t2t=t1+1H2,t1,t

D′t1Dt1

in thenumerator implies

B′t2 =βt2H1,t2(1− θ)D

′t2

Dt2+ βt1H1,t1ξB,t1


B′t1

−

∑t2t=t1+1 H2,t1,t


′t

Dt

)βt2H1,t2 +H2,t1,t2(1− θ)FB,t2

Ft2

B′t1

+βt2H1,t2(1− θ)


Dt1

)βt2H1,t2 +H2,t1,t2(1− θ)FB,t2

Ft2

M ′t1

−

∑t2t=t1+1H2,t1,t

(D′′tD′t− Dt′

Dt+ (1− θ)

(D′tDt− D′t1

Dt1

))βt2H1,t2 +H2,t1,t2(1− θ)FB,t2

Ft2

M ′t1

+(1− θ)

(βt2H2,t2 − βt1H1,t1 −

∑t2t=t1+1H2,t1,t

) D′t1Dt1


M ′t1.

Given that the carbon cycle ful�lls (2.32), the �rst-order conditionof the planner problem (use (2.34) in (2.58)) implies that the last termis zero. This produces

B′t2 =βt2H1,t2(1− θ)D

′t2

Dt2+ βt1H1,t1ξB,t1 −

∑t2t=t1+1H2,t1,t


′t

Dt

)βt2H1,t2 +H2,t1,t2(1− θ)FB,t2

Ft2

B′t1

+βt2H1,t2(1− θ)


Dt1

)βt2H1,t2 +H2,t1,t2(1− θ)FB,t2

Ft2

M ′t1

−

∑t2t=t1+1H2,t1,t

(D′′tD′t− Dt′

Dt+ (θ − 1)

(D′t1Dt1− D′t

Dt

))βt2H1,t2 +H2,t1,t2(1− θ)FB,t2

Ft2

M ′t1.

2.A.4 Calculations for model with elastic supply of

alternative-energy input

I will now calculate the e�ects of varying the exogenous variables whenthe supply of alternative energy is endogenous. Throughout, the Gs arede�ned as in (2.16). I will start by calculating the derivatives of theproduction function with respect to the endogenous variables and com-puting some combinations of derivatives that are useful in the subsequentcalculations. I will then calculate the derivative of H with respect to B.


After that, I will calculate the derivative of H with respect to each ofthe exogenous variables.

Derivatives with respect to endogenous variables

Using the fact that Y = AY Y and the derivatives (2.96)-(2.99) from2.A.1, the marginal products, in �nal goods production, of labor andthe alternative-energy input, respectively, are

FL =FGL

GE +GL

1

LY

FS =FGE

GE +GL

GS

GB +GS

1

S.

The equilibrium labor allocation condition (2.84) then gives

GL

GE +GL

1

LY=

GE

GE +GL

GS

GB +GS

aSS. (2.100)

A common term for all derivatives is 2aSFLS − FLL − a2SFSS. The

derivatives needed to calculate this expression are

FLS =FL1

σY

GE

GE +GL

GS

GB +GS

1

S

FLL =−FL1

σY

GE

GE +GL

1

LY

FSS =−FS[

1

σY

GL

GE +GL

GS

GB +GS

+1

σE

GB

GB +GS

]1

S.

Furthermore, the derivatives involving fossil-fuel use are

FB =FGE

GE +GL

GB

GB +GS

1

B

FBB =−FB[

1

σE

GS

GB +GS

+1

σY

GL

GE +GL

GB

GB +GS

]1

B

FBS =FB

[1

σE− 1

σY

GL

GL +GE

]GS

GB +GS

1

S

=FS

[1

σE− 1

σY

GL

GL +GE

]GB

GB +GS

1

B

FBL =FB1

σY

GL

GE +GL

1

LY

=FL1

σY

GE

GE +GL

GB

GB +GS

1

B.


Combining these derivatives and using (2.100) we obtain

FBB − θF 2B

F=−FB

1

σE

GS

GB +GS

1

B

−FB(

1

σY

GL

GE +GL

+ θGE

GE +GL

)GB

GB +GS

1

B

aSFBS − FBL =FL

(1

σE− 1

σY

)GB

GB +GS

1

B

=FB

(1

σE− 1

σY

)GS

GB +GS

aSS

2aSFLS − FLL − a2SFSS =FL

(1

σY

GS

GB +GS

+1

σE

GB

GB +GS

)aSS

+FL1

σY

1

LY. (2.101)

Calculation of ∂H∂B

From (2.88), we see that

1

u′(C)

∂H

∂B=FBB − θ

F 2B

F+

(FBSaS − FBL)2


.

Using the derivatives and expressions from the previous section, onearrives at

1

u′(C)

∂H

∂B=−FB

B

1σY σE

(1− GE

GE+GL

(GB

GB+GS

)2)

aSS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

−FBB

θσE

GEGE+GL

(GB

GB+GS

)2aSS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

(2.102)

−FBB

1σY

(1σE

GSGB+GS

+ θ GBGB+GS

)1LY(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

.

This expression is negative.

Calculation of ∂H∂AY

From (2.91), we have that

1

u′(C)

∂H

∂AY= FBAY − θ

FBFAYF

+(aSFBS − FBL) (aSFSAY − FLAY )


.


The derivatives of the production function with respect to AY are

FAY =F1

AY

FBAY =FB1

AY

FSAY =FS1

AY

FLAY =FL1

AY.

Combining the last two derivatives delivers

aSFSAY − FLAY = (aSFS − FL)1

AY= {(2.84)} = 0

This implies that the reallocation of labor satis�es

LSAY =aSFSAY − FLAY


= 0

The �rst two derivatives then yield

1

u′(C)

∂H

∂AY= FBAY − θ

FBFAYF

= (1− θ)FB1

AY

Calculation of ∂H∂AL

From (2.91), we obtain

1

u′(C)

∂H

∂AL= FBAL − θ

FBFALF

+(aSFBS − FBL) (aSFSAL − FLAL)


.

The derivatives of the production function with respect to AL are

FAL =FGL

GE +GL

1

AL

FBAL =FB1

σY

GL

GE +GL

1

AL

FSAL =FS1

σY

GL

GE +GL

1

AL

FLAL =FL

(1− 1

σY

GE

GE +GL

)1

AL.


Combining the �rst two derivatives yields

FBAL − θFBFALF

=FB1

σY

GL

GE +GL

1

AL− θFB

GL

GE +GL

1

AL

=FB

(1

σY− θ)

GL

GE +GL

1

AL.


aSFSAL − FLAL = aSFS1

σY

GL

GE +GL

1

AL− FL

(1− 1

σY

GE

GE +GL

)1

AL

=FL1− σYσY

1

AL.

This delivers a reallocation of labor satisfying

LSAL =aSFSAL − FLAL


=FL

1−σYσY

1AL


,

which is positive under assumption (2.13).The change in the marginal value of fossil-fuel use is

1

u′(C)

∂H

∂AL=FBAL − θ

FBFALF

+(aSFBS − FBL) (aSFSAL − FLAL)


=

FB

(1σY− θ)

GLGE+GL

1ALFL

(1σY

GS+ 1σE

GB

GB+GS

aSS

+ 1σY

1L

)FL

(1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ FL1σY

1LY

+FB

(1σE− 1

σY

)GS

GB+GS

aSSFL

1−σYσY

1AL

FL

(1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ FL1σY

1LY

=FBAL

1σY

(1 + 1−σY

σE− θ)

GSGB+GS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

aSS.

+FBAL

(1σY− θ)

1σE

GLGE+GL

GBGB+GS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

aSS.

In general, the sign of this is expression is ambiguous. If θ ≤ 1 +1−σYσE

> 1 it is unambiguously positive. If θ > 1 + 1−σYσE

, the �rst termis negative while the second term is positive. When the supply of thealternative-energy input was inelastic, the marginal value of fossil-fueluse was increasing in AL under assumptions (2.13), (2.14) and (2.15).


Since aSFSAL −FLAL = FL1−σYσY

1AL

> 0, it follows from (2.89) that laboris moved from �nal good production to the production of the alternativeenergy input. This reallocation counteracts the positive e�ect on thevalue of fossil-fuel use and leaves us with an ambiguity.

Calculation of ∂H∂AE

From (2.91) we arrive at

1

u′(C)

∂H

∂AE= FBAE − θ

FBFAEF

+(aSFBS − FBL) (aSFSAE − FLAE)


.

The derivatives of the production function with respect to AE are

FAE =FGE

GE +GL

1

AE

FBAE =FB

(1− 1

σY

GL

GE +GL

)1

AE

FSAE =FS

(1− 1

σY

GL

GE +GL

)1

AE

FLAE =FL1

σY

GE

GE +GL

1

AE.

Combining the �rst two derivatives delivers

FBAE − θFBFAEF

=FB

(1− 1

σY

GL

GE +GL

)1

AE− θFB

GE

GE +GL

1

AE

=FB

(1− 1

σY

GL

GE +GL

− θ GE

GE +GL

)1

AE.

Combining the last two derivatives gives

aSFSAE − FLAE = aSFS

(1− 1

σY

GL

GE +GL

)1

AE− FL

1

σY

GE

GE +GL

1

AE

=FLσY − 1

σY

1

AE.

This produces a reallocation of labor of

LSAE =aSFSAE − FLAE


=FL

σY −1σY

1AE


,

which is negative under assumption (2.13).


The change in the marginal value of fossil-fuel use is

1

u′(C)

∂H

∂AE=FBAE − θ

FBFAEF

+(aSFBS − FBL) (aSFSAE − FLAE)


=FB

(1− 1

σY

GL

GE +GL

− θ GE

GE +GL

)1

AE

+FB

(1σE− 1

σY

)GS

GB+GS

aSSFL

σY −1σY

1AE

FL

(1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ FL1σY

1LY

=−FBAE

1σE

[1−σYσY

(GL

GE+GL

GBGB+GS

+ GSGB+GS

)]aSS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

−FBAE

θ−1σE

GEGE+GL

GBGB+GS

aSS

+ θ−1σY

1LY(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

.

This expression is negative under assumptions (2.13), (2.14) and (2.15).

Calculation of ∂H∂AB

From (2.91) we see that

1

u′(C)

∂H

∂AB= FBAB − θ

FBFABF

+(aSFBS − FBL) (aSFSAB − FLAB)


.

The derivatives of the production function with respect to AB are

FAB =FGE

GE +GL

GB

GB +GS

1

AB

FBAB =FB

(1− 1

σE

GS

GB +GS

− 1

σY

GL

GE +GL

GB

GB +GS

)1

AB

FSAB =FS

(1

σE− 1

σY

GL

GE +GL

)GB

GB +GS

1

AB

FLAB =FL1

σY

GE

GE +GL

GB

GB +GS

1

AB.


Combining the �rst two derivatives gives

FBAB − θFBFABF

=FB

(1− 1

σE

GS

GB +GS

− 1

σY

GL

GE +GL

GB

GB +GS

)1

AB

−θFBGE

GE +GL

GB

GB +GS

1

AB

=FBσE − 1

σE

GS

GB +GS

1

AB

−FB1− σYσY

GL

GE +GL

GB

GB +GS

1

AB

−(θ − 1)GE

GE +GL

GB

GB +GS

1

AB.


aSFSAB − FLAB = aSFS

(1

σE− 1

σY

GL

GE +GL

)GB

GB +GS

1

AB

−FL1

σY

GE

GE +GL

GB

GB +GS

1

AB

=FL

(1

σE− 1

σY

)GB

GB +GS

1

AB.

The reallocation of labor then becomes

LSAB =aSFSAE − FLAE


=FL

(1σE− 1

σY

)GB

GB+GS

1AB


,

which is negative under assumption (2.13).The change in the marginal value of fossil-fuel use is

1

u′(C)

∂H

∂AB=FBAB − θ

FBFABF

+(aSFBS − FBL) (aSFSAB − FLAB)


=FBAB

σE − 1

σE

GS

GB +GS

−FBAB

(1− σYσY

GL

GE +GL

+ (θ − 1)GE

GE +GL

)GB

GB +GS

FBAB

(1σE− 1

σY

)2GB

GB+GS

GSGB+GS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

aSS.

The �rst two terms give the same behavior as in the model withinelastic supply of the alternative-energy input. The fourth term is pos-itive. It captures the fact that an increase in AB reallocates labor from


provision of the alternative-energy input to �nal goods production. Notethat when either GB << GS or GS << GB, the fourth term is smalland the sign of ∂H

∂ABis the same as in the model with inelastic supply of

the alternative-energy input.

Calculation of ∂H∂AS

From (2.91), we note that

1

u′(C)

∂H

∂AS= FBAS − θ

FBFASF

+(aSFBS − FBL) (aSFSAS − FLAS)


.

The derivatives of the production function with respect to AS are

FAS =FGE

GE +GL

GS

GB +GS

1

AS

FBAS =FB

(1

σE− 1

σY

GL

GE +GL

)GS

GB +GS

1

AS

FSAS =FS

(1− 1

σE

GB

GB +GS

− 1

σY

GL

GE +GL

GS

GB +GS

)1

AS

FLAS =FL1

σY

GE

GE +GL

GS

GB +GS

1

AS.

Combining the �rst two derivatives produces

FBAS − θFBFASF

=FB

(1

σE− 1

σY

GL

GE +GL

)GS

GB +GS

1

AS

−θFBGE

GE +GL

GS

GB +GS

1

AS

=FB

(1

σE− 1

σY

GL

GE +GL

− θ GE

GE +GL

)GS

GB +GS

1

AS.

Combining the last two derivatives yields

aSFSAS − FLAS = aSFS

(1− 1

σE

GB

GB +GS

− 1

σY

GL

GE +GL

GS

GB +GS

)1

AS

−FL1

σY

GE

GE +GL

GS

GB +GS

1

AS

=FL

(σE − 1

σE

GB

GB +GS

+σY − 1

σY

GS

GB +GS

)1

AS.

This implies that the reallocation of labor becomes

LSAS =aSFSAE − FLAE


=FL

(σE−1σE

GBGB+GS

+ σY −1σY

GSGB+GS

)1AS


,


the sign of which is ambiguous under assumption (2.13). It is positive ifGB >

σEσE−1

σY −1σY

GS and negative if GB <σEσE−1

σY −1σY

GS.

The change in the marginal value of fossil-fuel use is

1

u′(C)

∂H

∂AS=FBAS − θ

FSFASF

+(aSFBS − FBL) (aSFSAS − FLAS)


=−FBAS

GS

GB +GS

(1σY− 1

σE

)(1− 1

σE

GEGE+GL

GBGB+GS

)aSS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

−FBAS

GS

GB +GS

1σE

(θ − 1

σE

)GE

GE+GL

GBGB+GS

aSS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

−FBAS

GS

GB +GS

1σY

(θ − 1

σE

)1LY(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

This is negative under assumptions (2.13) and(2.14).

Calculation of ∂H∂aS

Using the expressions for FLS and FSS from 2.A.4 and the labor alloca-tion condition (2.84) we obtain

FS + aSLSFSS − LSFLS = FS

(σY − 1

σY

GS

GB +GS

+σE − 1

σE

GB

GB +GS

).

From (2.93), the reallocation of labor becomes

LSaS =FS + aSL

SFSS − LSFLS2aSFLS − a2

SFSS − FLL= FS

σY −1σY

GSGB+GS

+ σE−1σE

GBGB+GS


.

The sign of this expression is ambiguous under assumptions (2.13).

FBS − θFBFSF

= FB

(1

σE− 1

σY

GL

GE +GL

− θ GE

GE +GL

)1

S.


Using (2.94), we obtain

1

u′(F )

∂H

∂aS=

(FBS − θ

FBFSF

)LS

+

(FS + aSL

SFSS − LSFLS)

(aSFBS − FBL)


=−FBaS

(1σY− 1

σE

)(1− 1

σE

GEGL+GE

GBGB+GS

)aSS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

GS

GB +GS

−FBaS

1σE

(θ − 1

σE

)GE

GL+GE

GBGB+GS

aSS(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

GS

GB +GS

−FBaS

1σY

(θ − 1

σE

)1LY(

1σY

GSGB+GS

+ 1σE

GBGB+GS

)aSS

+ 1σY

1LY

GS

GB +GS

.

This is negative under assumptions (2.13)-(2.15).

Chapter 3

The Role of the Nature of

Damages

3.1 Introduction

The emissions of CO2 from the burning of fossil fuels is believed to bean important driver behind climate change. Increasing concentrations ofgreenhouse gases in the atmosphere strengthens the green house e�ectand thereby increases the temperature on earth. With higher temper-ature follow many changes in the functioning of the earth system. TheIPCC assessment report (2007) on climate change describes a large rangeof e�ects of climate change. The discussed e�ects include a sea level rise,heat waves, storms, changes in disease vectors, agricultural productivi-ties and water availability.

From these examples of the expected e�ects, it can be seen thatclimate change will a�ect the economy in many di�erent ways. Whenbuilding integrated assessment models, with feedbacks between the cli-mate and the economy, the e�ects of a changing climate on the economymust be modeled explicitly and a decision must be made about exactlyhow the economy is a�ected. In this chapter I will investigate the con-sequences of di�erent possible choices.

Following Nordhaus' groundbreaking work, with the DICE/RICEmodels (see, e.g., Nordhaus and Boyer, 2000), the most common way ofmodeling these e�ects is to assume that they a�ect productivity. That is,that climate change a�ects the economy by changing - typically reducing- general TFP.

Other authors assume that climate variables instead enter directlyinto the utility function. Examples of this includes Acemoglu et al. 2012and van der Ploeg and Withagen (2012).

Weitzman (2010) investigates how the shape of the damage function

95

96 CHAPTER 3. THE ROLE OF THE NATURE OF DAMAGES

a�ects the results derived, concerning, e.g., optimal policy. He comparesthe e�ects of assuming that the climate state interacts multiplicativelyor additively with production capacity to provide welfare. He �nds thatthis choice is very important, especially if a calibrated model is extrapo-lated to situations with signi�cant climate change. While the analysis iscarried out using a utility function, the results also apply to a situationwhere damages enter into the production function.

Sterner and Persson (2008) argue that climate a�ects non-marketgoods and services. This can signi�cantly increase the estimates of thecosts related to future climate change induced damages if it is di�-cult to substitute the non-market goods and services with manufacturedgoods. Here the damages enter the production function, but not as thecommonly assumed multiplicative term. Their work can be seen as anillustration of the importance of the choice described by Weitzman.

In this chapter I will consider a di�erent alternative for how climatechange can a�ect the economy: through increased depreciation of capi-tal. Examples of e�ects that should a�ect depreciation of capital include�ooding and storms. As far as I know, no other papers consider the de-preciation rate as endogenous due to climate change. Greenwood et al.(1988) consider how endogenous capital depreciation, due to capacityutilization, a�ects the business cycle behavior.

In the models set up in this chapter, I will assume that climate af-fects productivity, the depreciation of capital and utility directly. I thenanalyze how the the di�erent types of damages a�ect the behavior ofthe model. The models I use are modi�ed neoclassical growth models.The factors of production are capital, labor, fossil-fuel based energy andalternative energy. The models build on the model of Golosov et al.(2011) which in turn builds on the models of Dasgupta and Heal (1974)and Stiglitz (1974).

The chapter can be seen as consisting of two parts. In the �rst part,I derive a formula for optimal taxation of fossil-fuel use in the contextof an in�nite time horizon model. In the second part of the chapter, Iuse a two-period model to study more in detail how the di�erent typesof damages a�ects the equilibrium fossil-fuel use.

The formula for the optimal tax that I derive, extends, under similarassumptions, the formula for the optimal tax from Golosov et al. (2011)to a situation where climate, in addition to a�ecting productivity, alsoa�ects utility directly and capital depreciation. Fossil-fuel use causesemissions of greenhouse gases which a�ects the future climate. Thee�ects on the climate are not internalized in an unregulated marketoutcome and taxation of fossil-fuel use can be used to correct for thisexternality. The derived formula gives the optimal per unit tax as a


constant times current total production in each period. The constant inthe tax formula adds up the e�ects of each of the three forms of damages.

In order to derive this formula, I assume that utility from consump-tion of manufactured goods is logarithmic in the consumed amounts,that the climate state is a linear function of past emissions, that (old)capital depreciates fully and that the consumption (and savings) rateis constant. Furthermore, I assume that the e�ect of the climate stateenters the utility function as an additive linear term and a�ects produc-tivity and depreciation as a multiplicative term with constant elasticity.Apart from these assumptions, the model is very general in terms of theinvolved functions and parameters.

The derived formula can be used as a guide when trying to combinedi�erent kinds of damages into a joint damage function (as is done in,e.g., Nordhaus' DICE/RICE models).

After that, in the context of a two-period model, I investigate how thedi�erent types of damages a�ect fossil-fuel use. In the two-period model,I treat climate as exogenous and consider how equilibrium fossil-fuel usedepends on productivity (in each of the periods), capital depreciationbetween the periods and on the climate state that enters the utilityfunction (in both periods).

In this setting, I consider two di�erent cases for the supply of fossilfuel. The �rst case is the oil case, where there are no extraction costsfor fossil fuel and where the constraint on the total amount of availablefossil fuel always binds. This can also be seen as the Hotelling case,where fossil-fuel use is completely driven by the mechanisms studied inthe classic work by Hotelling (1931). In the second case, the coal case,scarcity does not matter at all, but extraction of fossil fuel uses capitaland labor. These extraction costs are assumed to be independent ofthe remaining stock. These two cases can be seen as two extremes withreality lying somewhere in between.

The conclusion from this analysis is that both the assumptions abouthow the climate a�ects the economy and the assumption about fossil-fuelsupply are very important for the e�ects of climate on equilibrium fossil-fuel use. One possible interpretation is that climate change increasescapital depreciation, decreases productivity in the future (in the secondperiod) or increases the marginal value of consumption of manufacturedgoods in the second period (due to a demand for adaptation measures).In this interpretation, climate e�ects will, within each of the cases, haveopposite e�ects if it is assumed to a�ect capital depreciation comparedto if it is assumed to a�ect productivity or utility. Comparing the twocases (oil and coal) the e�ects will, for each type of damage, be quitedi�erent.


I also make an attempt to assess whether the realization that therewill be climate related e�ects will amplify or dampen climate change. Todo this, I must determine what this means in each of the two cases. Inthe oil case, total fossil-fuel use is exogenously given and the division ofthe oil between the periods endogenously determined. I then interpretmore �rst-period fossil-fuel use as an ampli�cation of climate change. Inthe coal case, �rst-period fossil-fuel use is exogenously determined bythe resources available in the �rst period, while second-period fossil-fueluse is endogenously determined. I then interpret more fossil-fuel usein the second period as ampli�cation of climate change. Under thisinterpretation, climate e�ects on productivity or utility will dampenclimate change in the oil case and amplify it in the coal case. Conversely,climate e�ects on the depreciation of capital will amplify climate changein the oil case, at least if the supply of alternative energy is exogenouslygiven, but dampen it in the coal case.

The rest of the chapter is organized as follows. In the next sectionI set up the in�nite time horizon model. I solve it both for the plannersolution and for a decentralized equilibrium. I then derive the formula forthe optimal taxation and make some simplifying assumptions that givesthe tax formula a very simple form. After that, in section 3.3, I analyzehow the di�erent types of e�ects of climate change a�ect fossil-fuel usein the oil and coal case, respectively. The chapter is then concluded witha discussion of the results in section 3.4.

3.2 Model setup

The factors of production are capital K, labor L and energy E. Capi-tal is accumulated through investment while labor is exogenously given.Energy is a combination of fossil-fuel based and alternative energy. Gen-eration of both types of energy can require the use of inputs. The useof fossil fuel is associated with emission of CO2, which causes climatechange. The climate state, M , determined by the history of fossil-fueluse, a�ects the productivity of �nal goods and the depreciation of capitaland it also enters directly into the utility function.

Final goods production uses capital, labor and energy. It also de-pends on the climate state and on a technology factor AY . Let theamounts of inputs used in �nal goods production be KY , LY and EY ,respectively. I will further assume that the e�ect of the climate state onproduction enters as a multiplicative factor. Final goods production canthen be written

Y = DY (M)FY (AY , KY , LY , EY ), (3.1)

where DY gives the climate related e�ects on productivity, AY is a tech-

3.2. MODEL SETUP 99

nology factor and FY is increasing in all its arguments and has constantreturns to scale in K, L and E.1

Energy comes from the burning of fossil fuel, which are extractedfrom the ground, and from alternative-energy generation. Let the re-maining fossil-fuel resources at the beginning of period t be Qt ≥ 0 andthe amount extracted and burnt in period t be Bt ≥ 0. The remainingstock of fossil fuel and the fossil-fuel use in each period must ful�ll

Qt+1 = Qt −Bt ≥ 0 and Bt ≥ 0. (3.2)

Extraction may require the use of inputs. Let the amounts of inputs usedin fossil-fuel extraction be KB, LB and EB. The extraction technologycan also evolve over time. Let the state of the extraction technology bedenoted AB. Since the resources required for the extraction of fossil fuelmay depend on the amount of remaining resources (if, e.g., the cheapestresources are used up �rst, then more resources are required to extracta given amount of fuels the smaller are the remaining reserves), theproduction of fossil fuel also depend on Q. In principle, climate changecould also a�ect the extraction possibilities (for example by increasingthe available resources due to ice melting in the arctic or by directlya�ecting the productivity in the extraction sector). I will not take thee�ect of climate on the extraction possibilities into account here.

Combined, this implies that the available amount of fossil-fuel basedenergy in a period can be written as

B = FB(AB, KB, LB, EB, Q), (3.3)

where FB is increasing in all its arguments.Let the amount of alternative-energy use in a period be S. Let the

state of the alternative-energy generation technology be AS and the in-puts used in alternative-energy production be KS, LS and ES. These arethen combined into the alternative energy according to the productionfunction

S = FS(AS, KS, LS, ES), (3.4)

where AS is a technology factor and FS is increasing in all its argumentsand has constant returns to scale in K, L and E.

These two energy inputs are then combined into energy according tothe production function

E = FE(AE, B, S), (3.5)

1I will sometimes refer to DY as damages. This can be a bit confusing since moredamages is associated with a lower value of DY . The same thing goes for the climatee�ect on depreciation, which is introduced below and denoted DK .


where AE is a technology factor and FE is increasing in all its argumentsand has constant returns to scale in B and S.

I will assume that total labor is exogenously given and equal to L.The climate state depends on the past history of fossil-fuel use:

Mt = M(Bt, Bt−1, . . . , B0). (3.6)

The dependency of climate on the history of emissions is, in reality, verycomplicated. The resulting increase in the concentration of CO2 in theatmosphere, following emissions from the burning of fossil fuel, dependson the working of the carbon cycle. Increased concentrations of CO2 inthe atmosphere then changes the radiative balance leading to an increasein the temperature.

Below, I will make the assumption that the marginal e�ect of fossil-fuel use in period t on the climate state in period t′ ≥ t only depends ont′ − t. This can be described by

dMt′

dBt

=

{φt′−t if t′ ≥ t0 if t′ < t

. (3.7)

While this ignores very much of the complexity of the climate system,it can be shown (see Golosov et al., 2011) that a model of this kind canreplicate the behavior of the climate system in the DICE/RICE modelswell.

Capital is depreciated here for two reasons. Firstly, there is standarddepreciation of capital given by a depreciation factor δ. This leaves theresources that can be used for consumption or investment

Yt + (1− δ)Kt = Ct + It.

The investments are then depreciated further by climate change in-duced e�ects. This depreciation is given by the factor DK,t = DK(Mt)and next period capital is

Kt+1 = ItDK,t ⇒ It =Kt+1

DK,t

. (3.8)

Consumption can then be written

Ct = Yt + (1− δ)Kt −Kt+1

DK(Mt). (3.9)

In each period t, the inputs are divided between �nal goods produc-tion and production of fossil-fuel based or alternative energy. This givesthe conditions

Kt =KB,t +KS,t +KY,t

Lt =LB,t + LS,t + LY,tEt =EB,t + ES,t + EY,t

(3.10)

3.2. MODEL SETUP 101

that must hold for all t. I will assume that resources can be reallocatedfreely between periods so that the allocation of K, L and E is a staticdecision made in each period.

In each period t, utility derived in that period depends on the amountof consumption Ct and on the climate state Mt. Total utility derivedover the in�nite time horizon is

∞∑t=0

βtU(Ct,Mt). (3.11)

Thus, the climate related e�ects appear in three di�erent places in themodel: in productivity (DY ), in capital depreciation (DK) and in theutility function. While moderate climate change could potentially bebene�cial in some places, the reasonable assumption is that, in a globalmodel, climate change has negative consequences. This implies thederivative signs

D′Y (M) ≤ 0, D′K(M) ≤ 0 and UM(C,M) ≤ 0.

The utility function also captures how climate change a�ects themarginal utility of consumption of the manufactured good. If the manu-factured good can be used to adapt to adverse e�ects of climate change,the reasonable sign of the cross derivative is

UCM ≥ 0

This completes the description of the model. I will now solve themodel both for the planner solution and for a decentralized equilibriumwith taxation of fossil fuel. Since there are climate externalities, theplanner solution and the decentralized equilibrium will, in general, notcoincide and there will be scope for improving on the decentralized equi-librium using taxation.


The planner is assumed to want to maximize welfare and be restrictedonly by the physical constraints on the economy. This means that theplanner wants to maximize (3.11) given all the conditions stated above.The set of variables that the planner chooses is{

Yt, Ct, Kt+1, Et, Bt, St, {KX,t, LX,t, EX,t}X∈{B,S,Y } , Qt+1,Mt

}∞t=0

.

(3.12)In principle, all variables should be subject to non-negativity con-

straints. Except for the case of Qt, I will assume that the non-negativity


constraints never bind. I will therefore exclude these from the formula-tion of the problem. This presumption would, for example, be satis�edif all the production functions ful�lled Inada conditions with respect toall inputs, that is, if the marginal product of an input, in all sectors,goes to in�nity when use of the input goes to zero. In the example insection 3.2.5 I will assume that energy production does not use capital.That means that the Inada conditions are not ful�lled there, but it isstraightforward to adapt the equilibrium conditions to that case.

I will not set up the planner problem explicitly. Instead I will writedown the Lagrangian of the problem, which is given by

L=∞∑t=0

βtU(Ct,Mt) +∞∑t=0

λC,t

[Yt + (1− δ)Kt −

Kt+1

DK(Mt)− Ct

]+∞∑t=0

λY,t [DY (Mt)FY (AY,t, KY,t, LY,t, EY,t)− Yt]

+∞∑t=0

λE,t [FE(AE,t, Bt, St)− Et]

+∞∑t=0

λB,t [FB(AB,t, KB,t, LB,t, EB,t, Qt)−Bt]

+∞∑t=0

λS,t [FS(AS,t, KS,t, LS,t, ES,t)− St]

+∞∑t=0

∑Z∈{K,L,E}

µZ,t

Zt − ∑X∈{B,S,Y }

ZX,t

∞∑t=0

λM,t [Mt(Bt, Bt−1, . . . , B0)−Mt]

∞∑t=0

λQ,t [Qt+1 +Bt −Qt] +∞∑t=0

µQ,tQt,

where all the µs and λs are multipliers.

Taking �rst-order conditions with respect to each of the variablesgives


Yt :λC,t = λY,t (3.13)

Ct :λC,t = βtUC,t (3.14)

Kt+1 :λC,t = DK(Mt) [λC,t+1(1− δ) + µK,t+1] (3.15)

Et :λE,t = µE,t (3.16)

Bt :λB,t = λE,tFB,E,t + λQ,t +∞∑t′=t

λM,tdMt′

dBt

(3.17)

St :λS,t = λE,tFE,S,t

Qt+1 :λQ,t+1 = λQ,t + λB,t+1FB,Q,t+1 + µQ,t+1 (3.18)

Mt :λM,t = βtUM,t + λC,tKt+1

DK(Mt)

D′(Mt)

DK(Mt)+ λY,tYt

D′Y (Mt)

DY (Mt).(3.19)

For Z ∈ {K,L,E}, the �rst order conditions with respect to ZY,t,ZB,t and ZS,t give that

µZ,t = λY,tDY (Mt)FY,Z,t = λB,tFB,Z,t = λS,tFB,S,t. (3.20)

These expressions state that each of the inputs capital, labor and energyshould be allocated across sectors so that their marginal values in eachsector are equalized.

Combining (3.14) and (3.15) gives the capital accumulation condition

βtUC,t =[µK,t+1 + βt+1UC,t+1(1− δ)

]DK(Mt).

This equation is similar to the standard Euler equation derived in a one-sector model. There are two di�erences. The �rst di�erence is the factorDK(Mt), which captures that a share of any investment is lost due toclimate related damages. This lowers the returns to investment. Thesecond di�erence is the term µK,t+1 appearing where there is usually amarginal product of capital in period t + 1. From equation (3.20) inperiod t+ 1, and with Z = K, it follows that

µK,t+1 = λY,t+1DY (Mt+1)FY,K,t+1 = λB,t+1FB,K,t+1 = λS,t+1FS,K,t+1,

where each of these expressions is the shadow value of the output inthe sector times the marginal product of capital in that sector. Thatis, µK,t+1 gives the marginal value of the marginal product of capital ineach of the sectors.

From (3.13) and (3.14), the shadow value of �nal goods productionin period t + 1 is λY,t+1 = βt+1UC,t+1. Substituting this expression intothe capital accumulation condition gives

βtUC,t = βt+1UC,t+1 [DY (Mt+1)FY,K,t+1 + 1− δ]DK(Mt). (3.21)


Turning now, instead, to the fossil-fuel use decision, the shadow valueof fossil-fuel use in period t is λB,t and it is given in (3.17). It can bedivided into three parts. The �rst term consists of the shadow value ofenergy times the marginal product of fossil fuel in producing energy: themarginal value of using fossil fuel in production. The second term is theshadow value of the remaining stock. From (3.18) it can be seen that thisshadow value captures the scarcity value of the resource (coming fromthe �niteness of the resource) and the dependency of the extraction costson the remaining stock. Each of these factors gives a value to keepingfuels in the ground. The third term in (3.17) captures the marginal costof the climate damages caused from period t and onwards caused byemissions in period t. It is the sum, from t to in�nity, of the shadowvalue of the climate state in that period, times the e�ect on the climatestate of the emissions in period t.

Substituting for λY,t and λC,t from (3.13) and (3.14), respectively,and for Kt+1

DK(Mt)from (3.8) into (3.19), the shadow value of the climate

state becomes

λM,t′ = βt′[UM,t′ + UC,t′

(It′D′K(Mt′)

DK(Mt′)+ Yt′

D′Y (Mt′)

DY (Mt′)

)].

The right-hand side captures the three e�ects that the climate isassumed to have. The �rst term captures the direct e�ect on welfare,the second term captures the e�ect on the capital stock and the thirdterm captures the e�ect on productivity. The e�ects are computed asmarginal utility e�ects in period t′ discounted back to period 0.

Using (3.20), with Z = E, and (3.16), the shadow value of energy is

λE,t = λY,tDY (Mt)FY,E,t = {(3.14) and (3.15)} = βtUC,tDY (Mt)FY,E,t.

Substituting for λM,t′ and λE,t in (3.17) gives the fossil-fuel use condition

λB,t =∞∑t′=t

βt′[UM,t′ + UC,t′

(It′D′K(Mt′)

DK(Mt′)+ Yt′

D′Y (Mt′)

DY (Mt′)

)]dBt

+βtUC,tDY (Mt)FY,E,tFE,B,t + λQ,t. (3.22)

This concludes the characterization of the solution to the plannerproblem. The two most important conditions are the capital accumula-tion condition (3.21) and the fossil-fuel use condition (3.22).

3.2.2 Decentralized equilibrium with taxation

In the decentralized equilibrium, the decisions are made by di�erentactors. Households are assumed to derive income from renting out cap-ital to �rms, from renting out labor to �rms, from pro�ts in �rms that


they own shares in and from government lump-sum transfers. There arefour di�erent types of �rms in the model: �nal goods producing �rms,fossil-fuel extracting �rms, alternative-energy producing �rms and �rmsthat combine the output from the two energy sources into the compos-ite energy good. Let the pro�ts of these �rms be πY , πB, πS and πErespectively. I will assume that all �rms are owned in equal shares byall households.

The government taxes sales of fossil fuel with a per unit tax τ andpays lump-sum transfers g to the households.

I will assume that all agents act as price takers. I will normalize theprice of �nal goods in each period to 1. In addition to this, there will be5 prices. Let the energy price be pE, the price of labor be w, the rentalrate of capital be r, the price of fossil-fuel based energy be pB and theprice of alternative energy be pS.

A competitive equilibrium can now be de�ned as a sequence of prices{pE,t, wt, rt, pB,t, pS,t}∞t=0, quantities (3.12) and taxes and governmenttransfers {τt, gt}∞t=0 such that

• All �rms maximize pro�ts

• Households maximize utility

• The government budget is balanced in each period

• The climate state is given by (3.6)

I will �rst solve the �rms' pro�t maximization problems and thenthe households' utility maximization problem. After that, I will combinethese conditions, and the government's balanced budget constraint, tocharacterize the equilibrium allocation. I will not explicitly need to usethe equation for the climate state. In this section, I will treat the taxesas given. In section 3.2.3 I will then derive the taxes that implement theplanner solution.

Firms

Final good producing �rms, alternative-energy producing �rms and com-posite energy producing �rms all rent or buy their inputs in each periodand therefore face a static problem in each period. Under assumptionsof perfect competition and constant returns to scale (in capital, labor,and energy for �nal goods and alternative-energy producing �rms, andto fossil-fuel based and alternative energy for composite energy produc-ing �rms), each �rm will use each input up to the point where the valueof the marginal product of the input equals its price. This gives the


equilibrium conditions2

rt =DY (Mt)FY,K,t = pS,tFS,K,t (3.23)

wt =DY (Mt)FY,L,t = pS,tFS,L,t (3.24)

pE,t =DY (Mt)FY,E,t = pS,tFS,E,t (3.25)

pB,t = pE,tFE,B,t (3.26)

pS,t = pE,tFE,S,t.

Under the constant returns to scale assumptions, �nal goods, alter-native energy and composite energy producing �rms will all make zeropro�ts:

∀t : πY,t = πS,t = πE,t = 0.

The fossil-fuel extracting �rms own fossil-fuel resources and thereforemust take the e�ect of extraction on the remaining stock into account.This makes their pro�t maximization problem dynamic. They choosean extraction path to maximize the discounted sum of pro�ts.

The pro�t made, in each period, is the after tax revenues from thesales of fossil fuel minus the costs for the inputs used in extraction:

πB,t = (pB,t − τt)Bt − rtKB,t − wtLB,t − pE,tEB,t. (3.27)

The discount rate between two periods is given by the net return toinvestment in capital, that is, the return that pro�ts invested in capitalwould yield. De�ne the discount factors

Rt+1 = DK(Mt) [rt+1 + 1− δ] and Rt1,t2 =

{1 if t1 = t2∏t2

t=t1+1Rt if t2 > t2,

where Rt+1 gives the return in period t + 1 from investments made inperiod t and Rt1,t2 gives the discount factor in period t1 for incomederived in period t2 ≥ t1.

The fossil-fuel extracting �rms are thus assumed to maximize thediscounted pro�t stream. For simplicity, I will assume that the fossil-fuel resources consists of large number of identical sources owned bydi�erent �rms. Each �rm solves the problem

max∞∑t=0

1

R0,t

[(pB,t − τt)Bt − rtKB,t − wtLB,t − pE,tEB,t]

s.t. ∀t : Bt = FB(AB,t, KB,t, LB,t, EB,t, Qt)

Qt+1 = Qt −Bt ≥ 0; Bt ≥ 0,

2As in the case of the planner solution, I will assume interior solutions.


where the maximization is over {Bt, Qt+1, KB,t, LB,t, EB,t}∞t=0. The La-grangian of this problem is

L=∞∑t=0

1

R0,t

[(pB,t − τt)Bt − rtKB,t − wtLB,t − pE,tEB,t]

+∞∑t=0

λB,t [FB(AB,t, KB,t, LB,t, EB,t, Qt)−Bt]

+∞∑t=0

λQ,t [Qt+1 +Bt −Qt] +∞∑t=1

µQ,tQt.


Bt :λB,t =1

R0,t

(pB,t − τt) + λQ,t (3.28)

Qt+1 :λQ,t+1 = λQ,t + λB,t+1FB,Q,t+1 + µQ,t+1

KB,t :1

R0,t

rt = λB,tFB,K,t

LB,t :1

R0,t

wt = λB,tFB,L,t

EB,t :1

R0,t

pE,t = λB,tFB,E,t.

I have now derived the pro�t maximization condition for all types of�rms.

Households

In each period, households derive income from capital, labor, pro�tsfrom �rms they own shares in and government transfers. The only �rmsthat will make pro�ts are the fossil-fuel extracting �rms. Furthermore,I will not model trade in shares in �rms. In a representative householdmodel, there can, in equilibrium, be no trade in any shares. Includingtrade in shares would simply determine the equilibrium prices of thoseshares.

The households' budget constraint is

Ct +Kt+1

DK(Mt)= (rt + 1− δ)Kt + wtLt + πB,t + gt. (3.29)

Households solve the utility maximization problem

max{Ct,Kt+1}∞t=0

∞∑t=0

βtU(Ct,Mt)

s.t. (3.29) ∀t.



L=∞∑t=0

βtU(Ct,Mt)

+∞∑t=0

λC,t

[(rt + 1− δ)Kt + wtLt + πt + Tt − Ct −

Kt+1

DK(Mt)

].

The �rst-order conditions of this problem are

Ct :λC,t = βtUC,t

Kt+1 :λC,t = λC,t+1 [rt+1 + 1− δ]DK(Mt).

Combining these two conditions delivers the households' Euler equation:

UC,tβUC,t+1

= DK(Mt) [rt+1 + 1− δ] . (3.30)

This concludes the solution of the households' utility maximizationproblem.

Government

In each period, the government receives tax revenues τtBt and pays lump-sum transfers gt. The requirement that the government's budget mustbe balanced in each period gives that

gt = τtBt. (3.31)

Allocation

I have now derived all the equilibrium conditions and will combine themto characterize the equilibrium allocation. In particular, I will derive thecapital accumulation and fossil-fuel use conditions. I will also verify thatall income in the economy goes to the households so that the households'budget constraint, in equilibrium, coincides with the resource constraintof the economy.

Substituting for the price of capital (3.23) in the Euler equation (3.30)gives

UC,t = βUC,t+1DK(Mt) [DY (Mt+1)FY,K,t+1 + 1− δ] . (3.32)

Comparing this to condition (3.21), they are the same.Substituting (3.25) into (3.26), the fossil-fuel price in period t be-

comespB,t = pE,tFE,B,t = DY (Mt)FY,E,tFE,B,t. (3.33)


Using the households' Euler equation (3.30), the discount factor is

R0,t =t∏

t′=1

Dk(Mt′−1) [rt′ + 1− δ] =t∏

t′=1

UC,t′−1

βUC,t′=

UC,0βtUC,t

. (3.34)

Substituting the fossil-fuel price (3.33) and the discount factor (3.34)into the fossil-fuel owners' �rst-order condition with respect to Bt (3.28)and rewriting gives

UC,0λB,t = βtUC,tDY (Mt)FY,E,tFE,B,t − βtUC,tτt + UC,0λQ,t. (3.35)

Comparing this equation to the fossil-fuel use condition from the plan-ner solution (3.22), they are in some ways similar, but there are someimportant di�erences. The e�ect of fossil-fuel use on the climate is notat all internalized here. In the place of the value of fossil-fuel use inproduction there is now the net of tax pro�t from fossil-fuel use. A lessimportant di�erence is that the multipliers are here multiplied by UC,0.Apart from that, the shadow value of the stock of fossil fuel enters thecondition in much the same way as in (3.22). The comparison between(3.22) and (3.35) is what will give the expression for the optimal tax insection 3.2.3.

I will now verify that all income goes to the households. Substitutingfor the pro�t from the fossil-fuel extracting �rms (3.27) and the lump-sum transfer (3.31) in the households' budget constraint (3.29) gives

Ct +Kt+1

DK(Mt)= (rt + 1− δ)Kt + wtLt

+ (pB,t − τt)Bt − rtKB,t − wtLB,t − pE,tEB,t + τtBt

= (Kt −KB,t) rt + (Lt − LB,t)wt + pB,tBt

−pE,tEB,t + (1− δ)Kt.

Using that alternative-energy generating �rms make zero pro�ts, weobtain

pS,tSt = rtKS,t + wtLS,t + pE,tES,t.

Similarly, using the fact that composite energy producing �rms makezero pro�ts, one arrives at

pE,tEt = pS,tSt + pB,tBt

= rtKS,t + wtLS,t + pE,tES,t + pB,tBt.

This delivers

pB,tBt = pE,tEt − rtKS,t − wtLS,t − pE,tES,t.


Substituting this into the households' budget constraint gives

Ct +Kt+1

DK(Mt)= (Kt −KB,t) rt + (Lt − LB,t)wt

+pE,tEt − rtKS,t − wtLS,t − pE,tES,t−pE,tEB,t + (1− δ)Kt

= (Kt −KB,t −KS,t) rt + (Lt − LB,t − LS,t)wt+ (Et − EB,t − ES,t) pE,t + (1− δ)Kt

=KY,trt + LY,twt + EY,tpE,t + (1− δ)Kt

= {(3.23), (3.24) and (3.25)}=DY (Mt) (FY,K,tKY,t + FY,L,tLY,t + FY,E,tKE,t)

+(1− δ)Kt

=DY (Mt)FY,t + (1− δ)Kt

=Yt + (1− δ)Kt.

The resulting equation is the resource constraint from the plannersolution (3.9).

This concludes the characterization of the decentralized equilibrium.The most important conditions are the capital accumulation condition(3.32) and fossil-fuel use condition (3.35).


I will now derive a formula for taxes that, in the decentralized equilib-rium with taxation, allows implementation of the planner solution.

As noted above, comparing the Euler equations from the plannersolution (3.21) and the decentralized equilibrium (3.32) they are thesame. Comparing the fossil-fuel use conditions (3.22) and (3.35), thetaxes {τt}∞t=0 should be chosen to make the two conditions equivalent.This can be achieved by in each period t choosing the per-unit tax

τt = −∞∑t′=t

[βt′UC,t′

βtUC,t

(It′D′K(Mt′)

DK(Mt′)+ Yt′

D′Y (Mt′)

DY (Mt′)

)+βt′UM,t′

βtUC,t

]dMt′

dBt

,

(3.36)all evaluated at the allocation from the planner solution.

It can also be shown that all the conditions for intratemporal allo-cation of resources between �nal goods production, fossil-fuel extractionand alternative-energy generation are the same in both cases.

So, if the tax is set according to this formula, the conditions forthe planner solution and the decentralized equilibrium are equivalent.Therefore, the taxes given by (3.36) implement the planner solution.The expression within the square bracket in (3.36) is the sum of the


three types of marginal damages in period t′, caused by a change in theclimate state, Mt′ , expressed in terms of period t consumption. This isthen multiplied by the marginal e�ect of emissions in period t on theclimate state in period t′ and summed over t′ ≥ t. Thus, the right-handside (3.36) gives total discounted marginal damages caused by emissionsin period t.

3.2.4 Simplifying assumptions

I will now make some simplifying assumptions that allow me to derive avery simple formula for the tax.

The utility function will be assumed to have the following form:

U(C,M) = ln(C)− κUM. (3.37)

I assume that the damages to the capital stock and to productivity havethe following forms:

DY (M) = e−κYM and DK(M) = e−κKM . (3.38)

I also assume that the carbon cycle is linear so that the e�ect of emissionsin period t, on the future state of the climate, is given by (3.7). As shownby Golosov et al. (2011), this speci�cation, with climate only a�ectingproductivity, can replicate the climate systems of the DICE/RICE mod-els well.

Under these assumptions, the optimal per unit tax becomes

τt = Ct

∞∑t′=t

βt′−t[It′

Ct′κK +

Yt′

Ct′κY + κU

]φt′−t.

The tax can be rewritten using the consumption rate

ct =CtYt.

Using the consumption rate, the ratio of investment to consumption is

Ct + It = Yt + (1− δ)Kt ⇒ItCt

= YtCt− 1 + (1− δ)Kt

Ct= 1−ct

ct+ (1− δ) Kt

ctYt.

Assuming further that there is full depreciation (δ = 1)

ItCt

=1− ctct

.

The per unit tax then becomes

τt = ctYt

∞∑t′=t

βt′−t[

1− ct′ct′

κK +1

ct′κY + κU

]φt′−t. (3.39)


Assuming, �nally, that the consumption rate is constant (in the plan-ner solution), ct = c for all t, then the following proposition can bestated.

Proposition 3.1. Assume that the utility function and the damage func-tions are given by (3.37) and (3.38). Assume further that δ = 1, that thecarbon cycle ful�lls (3.7) and that the consumption rate c is constant.Then the per unit tax in period t should be

τt = Yt [(1− c)κK + κY + cκU ]∞∑t′=t

βt′−tφt′−t (3.40)

That is, the optimal tax is a constant times production in that period.

Proof. Follows from (3.39) with ct = c for all t.

So, in each time period, the per unit tax should be a (time-invariant)constant times production in that period. Comparing this formula tothe corresponding formula in Golosov et al. (2011) they are the sameif κK = κU = 0. If κK 6= 0 or κU 6= 0 the tax, unlike in Golosov etal., depends on the value of the consumption rate. Note also that Ihere need to assume full depreciation to get the tax as a constant timesproduction. The reason for this is that both the consumption and thesavings rate (as shares of production) need to be constant, which is onlypossible when there is full depreciation.

The consumption rate enters the calculations just made in two dif-ferent ways. Firstly, it matters for the marginal e�ects in period t′ andsecondly, it matters for the marginal utility of consumption in period t.Starting with period t′, the e�ects in terms of period t′ marginal utilityare given within the square brackets in equation (3.39). The amountof resources destroyed by the damages to capital depreciation is pro-portional to investment. The damages are expressed in utility termsthrough multiplication by the marginal utility of consumption leavingthe ratio 1−c

c. Both the numerator and denominator of this ratio im-

plies a negative dependency on c. The amount of resources destroyedby damages to production is proportional to production and these areexpressed in utility terms through multiplication by the marginal utilityof consumption resulting in the ratio 1

c. The damages to utility are al-

ready expressed in utility terms. When translating the e�ects in periodt′ into period t consumption, this is achieved through division by periodt marginal utility of consumption. This step results in the initial factorctYt in the right-hand side of (3.39). Thus, in relation to Yt this last stepgives a positive dependency of the tax on the consumption rate. The netdependencies of the di�erent terms in (3.40) on the consumption rate are


the net e�ects. This net e�ect is positive for damages to utility, zero fordamages to productivity (since the dependency on c from the two stepscancel) and negative for damages to depreciation.

3.2.5 A closed-form example

I will now provide an example where the conditions of proposition 3.1 areful�lled. I will assume that the utility function and the damage functionsare given by (3.37) and (3.38) and that there is full depreciation δ = 1.Furthermore, I will assume that capital is not used in either fossil-fuelextraction or in alternative-energy generation and that capital entersthe �nal goods production function as in a Cobb-Douglas productionfunction. The last two assumptions imply that

KY,t = Kt, KB,t = KS,t = 0 for all t

andYt = DY (Mt)FY,t = DY (Mt)FY,tK

αt ,

where α is a given constant and where FY,t may depend on all of thevariables that FY,t depends on except Kt.

With FY,t given like this, the marginal product of capital is

DY (Mt)FY,K,t = αDY (Mt)FY,tKα−1t = α

YtKt

.

Substituting this into the Euler equation of the planner solution(3.21) and using the fact that utility is logarithmic, we obtain

1

Ct= β

1

Ct+1

αYt+1

Kt+1

DK(Mt).

With full depreciation, period t investment is

It =Kt+1

DK(Mt).

The Euler equation can now be writtenItCt

= αβYt+1

Ct+1

.

This is solved, in each period, by setting

Ct = (1− αβ)Yt and It = αβYt.

That is, the consumption rate is constant and equal to 1− αβ. Usingproposition 3.1, the tax should, in each period, be given by

τt = Yt [αβκK + κY + (1− αβ)κU ]∞∑t′=t

βt′−tφt′−t.

So, the optimal tax can be calculated without actually solving for theoptimal path of fossil-fuel use and within-period allocation of inputs.


3.3 Two-period model

In this section, I will consider a two-period version of the model, to seehow changes in the variables a�ected by climate change (DK , DY andM) changes fossil-fuel use. I will treat the �climate� as exogenously givenand then derive the e�ects of varying the variables.

I will consider two di�erent cases for the extraction of fossil fuel. The�rst case will be the �oil� case where fossil fuel is costlessly extractedfrom a given total supply. The second case will be the �coal� case wherethe �niteness of the resource does not matter and where extraction usesresources.

The �rst period is the same as in the in�nite horizon model. Firstperiod production is Y1. This is divided between consumption and in-vestment. The �rst period capital K1 is given. This capital depreciatesat the rate δ. Investments made in period 1, and the remaining capi-tal, is subjected to climate related damages after which the share DK

remains.

K2 = (Y1 − C1 + (1− δ)K1)DK ⇒ C1 = Y1 + (1− δ)K1 −K2

DK

.

Second-period production is consumed:

C2 = Y2.

There is a climate related damage that a�ects productivity in eachperiod, DY,t, and a climate state, Mt, that a�ects utility directly.

Regarding the one-period utility function, I will now assume that, forgivenM , the utility function behaves as a standard CES-utility functionin the sense that

UCCUC

= −θ 1

C(3.41)

and that, as before,UCM ≥ 0

so that more climate change increases the marginal utility from con-suming manufactured goods. The particular utility function de�ned in(3.37) ful�lls the last condition with equality.

I will now also be speci�c about the production function. Capital andlabor is aggregated according to a Cobb-Douglas production functioninto a composite G

G = KαL1−α. (3.42)

Fossil fuel, B, and alternative energy, S, are combined, according toa CES production function with elasticity σE, into the composite energy

3.3. TWO-PERIOD MODEL 115

good E:

E = FE(B, S) =[γBB

σE−1

σE + γSSσE−1

σE

] σEσE−1

. (3.43)

The composites G and E are then aggregated into �nal goods ac-cording to a CES-function with elasticity σY :

Y = DY FY (G,E) = DY

[γGG

σY −1

σY + γEEσY −1

σY

] σYσY −1

. (3.44)

For notational simplicity, I do not include any productivities AY orAE here. Including those as multiplicative terms would not change theessence of the analysis that is to follow.

In the oil and coal cases below, I will make di�erent assumptionsabout how B and S are produced. I will treat the climate as exogenous.I will then vary the climate related variables to see how di�erent possiblee�ects of climate change will a�ect equilibrium fossil-fuel use.

The results will depend on the values of the elasticities in the utilityand production functions. I therefore need to make some assumptionsabout those parameters.

Above I assumed that utility was logarithmic, that is, θ = 1. This islikely to be a reasonable assumption; Layard et al. (2008) �nd a valueof θ between 1.2 and 1.3.

There is low substitutability between energy and other inputs in theshort run. Hassler et al. (2011) estimate the elasticity to be 0.005. Thisimplies that σY should be small. The longer a time period is chosento be, the larger the substitutability can be expected to be since moreadjustments, e.g., more investments in energy e�ciency, are possible.Assuming limited substitutability still seems reasonable.

The substitutability between di�erent energy sources is high. If en-ergy is converted into electricity, the original source of the energy isirrelevant and there is perfect substitutability. For other energy uses,e.g., cars, investment in new machines may be required, but once theseare made, substitutability is high. In other uses, e.g., for �ying planes,fossil fuels seem much more di�cult to substitute away. Combined, thisseems to imply that, at the aggregate level, di�erent energy sources aregood, but not perfect, substitutes.

Based on all this, the following assumptions about parameters seemreasonable

0 <1

σE≤ 1 ≤ θ ≤ 1

σY<∞ (3.45)

3.3.1 The oil case

In this case, fossil fuel is costlessly extracted from a total supply Q. I willbelow �rst treat alternative energy as exogenously given. After that, I


will assume that alternative-energy generation requires labor and capital.The allocation of labor and capital between �nal goods production andalternative-energy generation will be a static decision that can be madewithin the period.

This means that production in each period depends on capital andamounts of fossil fuel used. That is, production can then be written as

Yt = DY,tFY (Bt, Kt).

Planner solution

Since I assume that the climate related parameters are exogenouslygiven, the planner solution and decentralized equilibrium will coincideand I will solve for the planner solution. All fossil fuel will be exhausted.Given parameter assumptions (3.45), fossil fuel will be used in both pe-riods and in the second period whatever amount was not used in the�rst period will now be used:

B2 = Q−B1.

This reduces the problem to choosing �rst-period fossil-fuel use andsecond-period capital. The planner problem is

maxB1,K2

U (DY,1FY (K1, B1) + (1− δ)K1 − K2

DK,M1

)+βU

(DY,2FY (K2, Q−B1),M2

) .The �rst-order conditions read

B1 :UC,1DY,1FY,B,1 = βUC,2DY,2FY,B,2

K2 :UC,11

DK

= βUC,2DY,2FY,K,2.

Substituting the second condition into the �rst condition gives the con-ditions

UC,1 = βDKUC,2DY,2FY,K,2 (3.46)

FY,B,2 =DKFY,K,2DY,1FY,B,1. (3.47)

These conditions completely characterize the solution.

E�ects of climate change

I will now vary the climate related variables to see how equilibriumfossil-fuel use reacts to these changes. Consider a change ∆ that a�ectsM1, M2, DK , DY,1 and DY,2. The equilibrium conditions (3.46) and(3.47) can be di�erentiated with respect to ∆ to identify the e�ect on


the equilibrium choice of investment and allocation of fossil-fuel use. Iwill let primes denote full derivatives with respect to ∆.

The derivatives of the production function are

Y ′1 =Y1

[D′Y,1DY,1

+FY,B,1

FY,1B′1

]

Y ′2 =Y2

[D′Y,2DY,2

+FY,B,2B

′2 + FY,K,2K

′2

FY,2

].

Di�erentiating both sides of condition (3.46) gives

d

d∆UC,1 =UC,1

[UCC,1UC,1

(Y ′1 −

K ′2DK

+K2

DK

D′KDK

)+UCM,1

UC,1M ′

1

]= {(3.41)}

=UC,1

[− θ

C1

(Y1

[D′Y,1DY,1

+FY,B,1

FY,1B′1

]− K ′2DK

+K2

DK

D′KDK

)]+UC,1

UCM,1

UC,1M ′

1

and

d

d∆βDKUC,2DY,2FY,K,2 = βDKUC,2DY,2FY,K,2

[UCC,2UC,2

Y ′2 +UCM,2

UC,2M ′

2

]+βDKUC,2DY,2FY,K,2

[D′KDK

+D′Y,2DY,2

]+βDKUC,2DY,2FY,K,2

[FY,KK,2

FY,K,2K ′2 +

FY,BK,2

FY,K,2B′2

]

Using βDKUC,2DY,2FY,K,2 = UC,1, C2 = Y2, B′2 = −B′1 and (3.41) deliv-


ers

d

d∆βDKUC,2DY,2FY,K,2 =UC,1

[−θ

(D′Y,2DY,2

− FY,B,2

FY,2B′1 +

FY,K,2

FY,2K ′2

)]

+UC,1

[UCM,2

UC,2M ′

2 +D′KDK

+D′Y,2DY,2

]+UC,1

[FY,KK,2

FY,K,2K ′2 −

FY,BK,2

FY,K,2B′1

]

=UC,1

[(1− θ)

D′Y,2DY,2

+D′KDK

+UCM,2

UC,2M ′

2

]+UC,1

(FY,KK,2

FY,K,2− θ FY,K,2

FY,2

)K ′2

+UC,1

(θFY,B,2

FY,2− FY,BK,2

FY,K,2

)B′1.

Setting

d

d∆UC,1 =

d

d∆βDKUC,2DY,2FY,K,2

and rewriting produces

K ′2 =

θC1Y1

D′Y,1DY,1

+(

1 + θC1

K2

DK

)D′KDK

+ (1− θ)D′Y,2

DY,2− UCM,1

UC,1M ′

1 +UCM,2UC,2

M ′2

θC1

1DK

+ θFY,K,2FY,2

− FY,KK,2FY,K,2

+

(θC1Y1

FY,B,1FY,1

+ θFY,B,2FY,2

− FY,BK,2FY,K,2

)θC1

1DK

+ θFY,K,2FY,2

− FY,KK,2FY,K,2

B′1. (3.48)

Turning, instead, to the Hotelling rule (3.47), we have that

dFY,B,2d∆

= FY,B,2

(FY,BK,2

FY,B,2K ′2 +

FY,BB,2

FY,B,2B′2

)

= FY,B,2

(FY,BK,2

FY,B,2K ′2 −

FY,BB,2

FY,B,2B′1

)


and that

d

d∆DKFY,K,2DY,1FY,B,1 =DKFY,K,2DY,1FY,B,1

[D′KDK

+D′Y,1DY,1

+FY,KK,2

FY,K,2K ′2

]

+DKFY,K,2DY,1FY,B,1

[FY,KB,2

FY,K,2B′2 +

FY,BB,1

FY,B,1B′1

]

= FY,B,2

[D′KDK

+D′Y,1DY,1

+FY,KK,2

FY,K,2K ′2

]

+FY,B,2

[FY,BB,1

FY,B,1− FY,KB,2

FY,K,2

]B′1.

SettingdFY,B,2d∆

=d

d∆DKFY,K,2DY,1FY,B,1,

using B′2 = −B′1 and rearranging then leads to

K ′2 =

FY,BB,1FY,B,1

+FY,BB,2FY,B,2

− FY,BK,2FY,K,2

FY,BK,2FY,B,2

− FY,KK,2FY,K,2

B′1 +

D′KDK

+D′Y,1DY,1

FY,BK,2FY,B,2

− FY,KK,2FY,K,2

.

Substituting this into (3.48) and rearranging gives that

ξBB′1 = ξDK

D′KDK

+ξDY,1D′Y,1DY,1

+(θ−1)D′Y,2DY,2

+UCM,1

UC,1M ′

1−UCM,2

UC,2M ′

2, (3.49)

where

ξDK =θFY,K,2FY,2

− FY,BK,2FY,B,2

+ θC1

(1K2−(FY,BK,2FY,B,2

− FY,KK,2FY,K,2

))K2

DK

FY,BK,2FY,B,2

− FY,KK,2FY,K,2

(3.50)

ξDY,1 =θFY,K,2FY,2

− FY,KK,2FY,K,2

+ θC1

[1DK− Y1

(FY,BK,2FY,B,2

− FY,KK,2FY,K,2

)]FY,BK,2FY,B,2

− FY,KK,2FY,K,2

.(3.51)

I show in appendix 3.B.2 that, for both of the two cases consideredbelow (exogenous and endogenous supply of alternative energy),

ξB > 0 (3.52)

By determining the signs of ξDK and ξDY,1 , I can use (3.49) to determinethe signs of the e�ects on �rst-period fossil-fuel use of varying DK , DY,1,DY,2, M1 and M2. I will do this for two di�erent cases, with exogenousand endogenous S respectively.


Alternative energy exogenously given

I will now assume that the amount of alternative energy in each period,S, is exogenously given. Under this assumption, capital is only used in�nal goods production and there are no indirect e�ects related to theredistribution of resources between �nal goods production and produc-tion of alternative energy. Then the derivatives of �nal goods productionwith respect to capital and fossil-fuel use are

FY,B =FY,EFE,B > 0

FY,K =FY,GGK > 0

FY,BK =FY,EGFE,BGK > 0

FY,BB =FY,EE (FE,B)2 + FY,EFE,BB < 0

FY,KK =FY,GG (GK)2 + FY,GGKK < 0.

In appendix 3.B.2 I show that these derivatives imply that ξB > 0.Turning to ξDK and ξDY,1 , it can be seen in (3.50) and (3.51) that

they contain the following three expressions that can be calculated usingthe derivatives of F and the derivatives of CES-production functions (seeappendix 3.A):

θFY,K

FY− FY,BK

FY,B= θ

FY,GGK

FY− FY,EGGK

FY,E=

(θ − 1

σY

)FY,GGK

FY(3.53)

θFY,K

FY− FY,KK

FY,K= θ

FY,GGK

FY− FY,GGG

2K + FY,GGKK

FY,GGK

= θFY,GG

FYα

1

K+

1

σY

FY,EFY

E

GαG

K+ (1− α)

1

K

=

[α

(θ − 1)FY,GG+ 1−σYσY

FY,EE

FY+ 1

]1

K(3.54)

FY,BK

FY,B− FY,KK

FY,K=FY,EGGK

FY,E− FY,GGGK

FY,K− GKK

GK

=1

σY

GK

G− GKK

GK

=1

σYα

1

K− (α− 1)

1

K=

(α

1− σYσY

+ 1

)1

K(3.55)

Substituting (3.53) and (3.55) into (3.50) delivers

ξDK = −

(1σY− θ)FY,G,2GK,2

FY,2+ θ

C1α 1−σY

σY

1DK(

α 1−σYσY

+ 1)

1K2

. (3.56)


Substituting (3.54) and (3.55) into (3.51) gives (after some calculations)

ξDY,1 = −θC1

(Y1 − K2

DK

)− 1 + α

(1σY− θ)FY,G,2G2

FY,2+ α

(θ Y1

C1− 1)

1−σYσY

α 1−σYσY

+ 1

(3.57)I can now state the following proposition:

Proposition 3.2. Assume that the parameters ful�ll (3.45) and thatUCM ≥ 0. Then

∂B1

∂DK

≤ 0,∂B1

∂DY,2

≥ 0,∂B1

∂M1

≥ 0,∂B1

∂M2

≤ 0

If, also, δ = 1, then∂B1

∂DY,1

≤ 0

Proof. Starting from (3.49) and using that ξB > 0, the results for DY,2,M1 andM2 follow from the parameter assumptions in (3.45) and UCM ≥0. The result for DK follows from (3.56) and the parameter assumptionsin (3.45). When δ = 1, Y1 = C1 + K2

DK. This implies that Y1

C1> 1 and

that θC1

(Y1 − K2

DK

)− 1 = θ − 1. The result for DY,1 now follows from

(3.57) and the parameter assumptions in (3.45).

The changes described in the proposition can be understood intu-itively. If DK decreases, second-period production and consumption lev-els go down and more consumption is redistributed to the second period.This redistribution can be achieved either by saving on fossil fuel or byinvesting in more capital. The decrease in DK decreases second-periodcapital, which increases the marginal product of capital but decreasesthe marginal product of fossil fuel. Therefore it is better to use morefossil fuel in the �rst period which allows for investing more in capital.So more fossil fuel is used in the �rst period to increase production. In-vestment also rises (and it should go up enough to decrease �rst-periodconsumption).

A decrease in DY,2 decreases the second-period levels of productionand consumption. This leads to an increase in second-period marginalutility from consumption. At the same time, the marginal products ofboth fossil fuel and capital in the second period decrease due to decreasedproductivity. If θ ≥ 1, the e�ect on the marginal utility of consumptiondominates the e�ect on the marginal products and both the �rst-periodlevel of investment and second-period fossil-fuel use increase. This leadsto a decrease in �rst-period fossil-fuel use.


An increase in M2 (assuming UCM ≥ 0) increase the marginal valueof consumption in the second period relative to the �rst period. Thisleads to a redistribution of resources from the �rst to the second period.This means that both �rst-period investment and second-period fossil-fuel use increases. This leads to a decrease in �rst-period fossil-fuel use.An increase in M1 works the other way around.

A decrease in DY,1 increases the marginal utility of consumption anddecreases the marginal product of fossil fuel in the �rst period. At thesame time �rst-period investments decrease, giving less second-periodcapital. This increases second-period marginal utility from consumptionand decreases second-period marginal product of fossil fuel relative tothe second-period marginal product of capital. If δ = 1, the entire netincome in period 1 comes from period 1 production and then more fossilfuel will be used in the �rst period to increase �rst-period consumptionand investment. If, instead, δ < 1, a share of �rst-period net incomecomes from non-depreciated capital. This weakens the e�ect on �rst-period consumption and second-period capital while the e�ect on �rst-period marginal product of fossil fuel is the same. This means thatit is no longer certain that the best thing is to compensate decreasedproductivity in the �rst period with more fossil-fuel use.

If climate change is interpreted as a decrease in DK and DY,2 andan increase in M2, then proposition 3.2 states that damages a�ectingproductivity and utility move emissions from the �rst period to the sec-ond period while damages a�ecting depreciation move emissions fromthe second period to the �rst period.

Alternative energy endogenously determined

I will now assume that alternative-energy generation is endogenouslydetermined. Generating alternative energy requires the use of labor andcapital. I will assume that the production function for alternative energyis such that the generated amount of energy is linear in the use of thecomposite G, de�ned in (3.42). I will also assume that the divisionof G into �nal good production and alternative-energy generation is astatic decision that can be made in each period. Let GY and GS bethe amounts of G used in �nal goods production and alternative-energygeneration, respectively. The amount of alternative energy is then

S = ASGS.

In the previous section, with alternative energy exogenously given,the change in fossil-fuel use caused by a change in �climate� was driven bytwo di�erent aspects. The �rst aspect was the redistribution of consump-tion between the periods. If the relative value of consumption increases


in one period the equilibrium changes in such a way that it redistributesconsumption towards that period. The second aspect was the relativeproductivity of the two factors (B and K) within the periods. The �rstaspect will be similar in this situation. The second aspect will be di�er-ent here. This is because the substitutability between the two factors isdi�erent when the alternative energy can be adjusted. In the productionfunction F , de�ned in (3.44), the substitutability between G and E islow. With exogenously given alternative energy, this translates into lowsubstitutability between B and K in F . In this section, the possibilityto use G to generate energy, increases the substitutability between Band K in F .

When considering changes in DY,2, M1 and M2 in the previous sec-tion, they were driven by consumption reallocation and the e�ects ofvarying these will therefore be the same here as there. This can also beseen in (3.49), where the factors in front of the changes in these climatevariables do not depend on the production function F but rather on theshape of the utility function.

When considering changes in DY,1 and DK , the e�ects depend onthe shape of the production function F . The factors multiplying themin (3.49), that is (3.50) and (3.51), depend on derivatives of F . Inparticular, they depend the following three combinations of derivatives:

θFY,K

FY− FY,BK

FY,B(3.58)

θFY,K

FY− FY,KK

FY,K(3.59)

FY,BK

FY,B− FY,KK

FY,K(3.60)

The reallocation of G between �nal-good production and alternative-energy generation must be taken into account when di�erentiating Fwith respect to B and K. I will �rst solve the problem of intratemporalallocation of G. This will allow me to calculate the needed derivativesof F with respect to B and K.

Becuase the division ofG between �nal goods production and alternative-energy generation is a static decision that can be made in each period,GY and GS can be seen as functions of B and G. The division of G isthus made to maximize production in the period. Production is givenby

DY FY (GY , FE (B,ASGS)) .


The intratemporal problem of dividing G can then be written

maxGY ,GS

DY FY (GY , FE (B,ASGS)) s.t. GY +GS = G

The �rst-order conditions for this problem give the optimality condition

FY,G = FY,EFE,SAS. (3.61)

This condition implicitly de�nes GY and GS as functions of B and G.This condition can be di�erentiated to derive the partial derivatives ofGY and GS with respect to B and G. This yields (see appendix 3.B.1,page 139)

GS,G =1σY

1GY

1σY

FYFY,EE

1GY

+ 1σE

FE,BB

FE

1GS

∈ (0, 1).

This expression reveals that as G is increased, GS is increased. Since thisderivative lies between zero and one, this implies that GY also increases.

The production function F can now be written

F (B,K) = FY (GY (B,G(K)), FE (B,ASGS(B,G(K)))) . (3.62)

This expression can be di�erentiated to give the partial derivatives (seeappendix 3.B.1, page 140)

FY,B =FY,EFE,B

FY,K =FY,GGK

FY,BK =FY,EGGKFE,B − FY,G(

1

σY− 1

σE

)FE,BE

GS,GGK

FY,KK =FY,GGG2K + FY,GGKK +

1

σYFY,G

1

GY

GS,GG2K .

The �rst-order derivatives are the same here as in the case withexogenous alternative energy. This is because the envelope theoremimplies that, since the allocation of G into GY and GS is chosen tomaximize production, the reallocation gives no �rst-order e�ects on F .The second-order derivatives consists of the same terms as in the casewith exogenous alternative energy. In addition to these, each derivativehas a term, of opposite sign, that captures the reallocation of G.

These derivatives can now be used to calculate expressions (3.58) and(3.60). The underlying calculations can be found in appendix 3.B.1 (seepage 143); here I only state some of the results.

Starting with expression (3.58) we can write

θFY,K

FY− FY,BKFY,B

=

(θ − 1

σY

)FY,GFY

GK+

(1

σY− 1

σE

)FY

FY,EEGS,G

FY,GFY

GK .


Comparing this equation to (3.53), we see that the �rst terms arethe same. The second term captures the redistribution of G that resultsfrom a change in G. The �rst term is negative and the second term ispositive, so that the new element here will tend to change the sign ofthe expression.

Substituting for GS,G and rewriting delivers the following equation(see appendix 3.B.1, page 143)

θFY,K

FY− FY,BK

FY,B=

(θ − 1

σE

)1σY

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

FY,GFY

GK

+

(θ(

1σY

FE,SS

FE+ 1

σE

FE,BB

FE

)− 1

σEσY

)GYGS

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

FY,GFY

GK .

This expression has the same sign as(θ − 1

σE

)1

σY+

(θ

(1

σY

FE,SS

FE+

1

σE

FE,BB

FE

)− 1

σEσY

)GY

GS

(3.63)

The �rst term here is positive. The sign of the second term is ambiguous.It will be positive if FE,SS >> FE,BB and negative if FE,SS << FE,BB.In order for (3.63) to be negative, FE,BB would have to be large enoughcompared to FE,SS and GY must be large enough relative to GS. Giventhat the energy sources are good substitutes, the income share of fossilfuel FE,BB will be large compared to the income share of alternativeenergy, FE,SS when a large amount of fossil fuel is used. Since energyand other inputs are poor substitutes, much of G will be used in �nalgood production if there is a signi�cant amount of energy. This impliesthat (3.63) will tend to be negative if fossil fuel is abundant and positiveotherwise.

Turning instead to expression (3.60), we see that it can be written(see appendix 3.B.1, page 143) as

FY,BK

FY,B− FY,KK

FY,K=

1

σY

1

GY

GK −GKK

GK

−[

1

σY

FYFY,EE

− 1

σE

FY,GGY

FY,EE

]1

GY

GS,GGK .

Here the �rst two terms are the same as in the previous section, see(3.55), and they are positive. The third term, captures the e�ects ofredistribution of G. As for expression just discussed, this term has theopposite sign, that is, it is negative. So, again, the e�ect of redistribution


of G will tend to change the sign of the e�ect compared to the previoussection.

Substituting for GS,G and rewriting gives (see appendix 3.B.1, page143)

FY,BK

FY,B− FY,KKFY,K

= α

1σY

1−σEσE

+(

1σY

1−σEσE

FE,SS

FE+ 1

σE

1−σYσY

FE,BB

FE

)GYGS

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

1

K+

1

K.

In the expression for ξDK , this shows up as

1−

(FY,BK

FY,B− FY,KK

FY,K

)K.

This expression will have the same sign as

1

σY

σE − 1

σE+

(1

σY

σE − 1

σE

FE,SS

FE+

1

σE

σY − 1

σY

FE,BB

FE

)GY

GS

. (3.64)

The �rst term here is positive. The sign of the second term depends onthe relative sizes of FE,SS and FE,BB. It is positive if FE,SS >> FE,BB,in which case (3.64) will be positive. If FE,BB >> FE,SS, the secondterm will be negative and the sign of (3.64) will depend on the relativesizes of the terms. The second term will be large if FE,BB is largerelative to FE,SS and if GY is large relative to GS. Thus, the analysisis the same here as for expression (3.63). This means that ξDK willbe positive if FE,B,2B2 is large compared to FE,S,2S2 and GY,2 is largecompared to GS,2.

Turning now to ξDY,1 (see (3.51)), its sign depends on (3.59) and[1DK− Y1

(FY,BK,2FY,B,2

− FY,KK,2FY,K,2

)]=

[K2DK

Y1−(FY,BK,2FY,B,2

− FY,KK,2FY,K,2

)K2

]Y1

K2

=

[K2DK−Y1

Y1+ 1−

(FY,BK,2FY,B,2

− FY,KK,2FY,K,2

)K2

]Y1

K2.

Apart from the �rst term in the parenthesis, this is the same expressionas for ξDK . The �rst term can be either positive or negative. If thereis full depreciation it is negative since the numerator then is investmentminus production, which must be negative. Expression (3.59) is positivesince FY,K > 0 and FY,KK < 0.

The conclusion from this discussion is that the e�ect of changes inDY,2, M1 and M2 will be the same here as in the case with exogenouslygiven alternative energy. The e�ect of changes in DK and DY,1, however,are ambiguous here.


3.3.2 The coal case

I will now instead consider the coal case. Here, there is no scarcity offossil fuel, by assumption; fossil fuel can be extracted using labor andcapital. I will assume that the composite input G = KαL1−α is usedin �nal goods production, fossil-fuel extraction and alternative-energygeneration. A more general formulation would be to specify di�erentproduction functions for di�erent sectors describing how labor and cap-ital are used together. Assuming that capital and labor enters the pro-duction function in the same way in all sectors simpli�es the analysisbut should not, if the di�erences are not too large, a�ect the resultssigni�cantly.

The composite input, G = KαL1−α, can be used in �nal goods pro-duction, extraction of fossil fuel or production of alternative energy. Thisallocation is made in each period and is (assuming free mobility of re-sources between sectors between time periods) a static decision.

I assume that both the extraction of fossil fuel and the alternative-energy generation are linear in G, so that

FB(GB) = ABGB and FS(GS) = ASGS. (3.65)

Under these assumptions, the planner solution can be divided intotwo separate subproblems. The �rst subproblem is the within-periodallocation of a given amount of G between the di�erent sectors. Thesecond subproblem is the intertemporal problem of how much, out of�rst-period production, to invest into second-period capital. I will solvethe two subproblems in turn.

Within-period allocation of G

I will now solve the problem of allocating a given amount of G in order tomaximize production. Let the amount of G used in �nal goods produc-tion, fossil-fuel extraction and alternative-energy generation be labeledGY , GB and GS, respectively. The problem of allocating G optimally isthen

maxGY ,GB ,GS

FY (GY , FE (FB (GB) , FS (GS))) s.t. GB +GS +GY = G.

There should also be non-negativity constraints on each of the Gs; how-ever, given the parameter assumptions in (3.45), these constraints willnever bind.


L = FY (GY , FE (FB (GB) , FS (GS))) + λ [G−GB −GS −GY ] .


The �rst-order conditions are

GB :λ = FY,EFE,BFB,G

GS :λ = FY,EFE,SFS,G

GY :λ = FY,G.

These �rst-order conditions express that the resulting marginal �nalgood product in all sectors should be the same. They can be rewrit-ten to give the equilibrium conditions

FE,SFS,G =FE,BFB,G (3.66)

FY,G =FY,EFE,BFB,G. (3.67)

For a given G, these conditions implicitly determine the allocation of Gbetween di�erent uses.

Using the production functions (3.43) and (3.65) and the derivativesof CES-production functions from appendix 3.A, condition (3.66) nowdelivers

ASγS

(FE

ASGS

) 1σE = ABγB

(FE

ABGB

) 1σE

⇒ GS =(ASAB

)σE−1 (γSγB

)σEGB.

Let

ξS =

(ASAB

)σE−1(γSγB

)σE> 0⇒ GS = ξSGB.

Substituting this expression into the production function for the en-ergy composite, we obtain

E =[γBB

σE−1

σE + γSSσE−1

σE

]=

[γBA

σE−1

σEB + γS (ASξS)

σE−1

σE

]GB.

Similarly, let

ξE =

[γBA

σE−1

σEB + γS (ASξS)

σE−1

σE

]> 0⇒ E = ξEGB.

Condition (3.67) then gives

γG

(FYGY

) 1σY

= γE

(FYE

) 1σY

γB

(FEB

) 1σE

AB

= γE

(FYξEGB

) 1σY

γB

(ξEGB

ABGB

) 1σE

AB

= γEγB

(FYGB

) 1σY

ξ1σE− 1σY

E AσEσEB .


Finally, let

ξY =

[γGγEγB

ξ1σY− 1σE

E A1−σEσE

B

]σY> 0⇒ GY = ξYGB

Summing up the di�erent uses of the composite G one then �nds that

G = GY +GB +GS = (1 + ξY + ξS)GB ⇒ GB =1

1 + ξY + ξSG

This in turn delivers

GY =ξY

1 + ξY + ξSG, GB =

1

1 + ξY + ξSG and GS =

ξS1 + ξY + ξS

G.

(3.68)Thus, the composite G is divided between the di�erent uses in propor-tions that are independent of G and also independent of DY,1, DY,2, DK ,M1 and M2. This also implies that fossil-fuel use is increasing in G.

Final goods production can now be written

Y =AY

[γGG

σY −1

σYY + γEE

σY −1

σY

] σYσY −1

= AY

[γGξ

σY −1

σYY + γEξ

σY −1

σYE

] σYσY −1

GB

=AY

[γGξ

σY −1

σYY + γEξ

σY −1

σYE

] σYσY −1 1

1 + ξY + ξSG.

Let

ξY G =

[γGξ

σY −1

σYY + γEξ

σY −1

σYE

] σYσY −1 1

1 + ξY + ξS,

givingY = AY ξY GG. (3.69)

For the results derived below, there are two intratemporal implica-tions that are particularly interesting. The �rst of these is that, as canbe seen in (3.68), the amount of fossil fuel extracted in a period is strictlyincreasing in G and therefore also in the amount of capital. The secondresult is that production in a period can be written as a factor, whichis independent of G, DY,1, DY,2, DK , M1 and M2, times G. This meansthat production in a period, as a function of capital, inherits many ofthe properties of G as a function of K. In particular,

∂Y

∂K> 0 and

∂2Y

∂K2< 0. (3.70)


Intertemporal allocation

Since the allocation of G between the di�erent sectors is a static decision,that can be made in each period, production in each period can bewritten as a function of only capital

Yt = DY,tF (Kt).

The planner problem is therefore

maxK2

U (DY,1F (K1) + (1− δ)K1 − K2

DK,M1

)+βU

(DY,2F (K2) ,M2

) The �rst-order condition with respect to K2 can be stated as

UC,1 = βDKUC,2DY,2F′(K2).

Consider now a change ∆ that a�ects DK , DY,1, DY,2, M1 and M2.Let primes denote full derivatives with respect to ∆. Then

dUC,1d∆

=UCC,1

[Y1

D′Y,1DY,1

− K ′2DK

+K2

DK

D′KDK

]+ UCM,1M

′1

d

d∆βDKUC,2DY,2F

′(K2) =UC,1

[UCC,2UC,2

(Y2

D′Y,2DY,2

+DY,2F′(K2)K ′2

)]+UC,1

[D′KDK

+D′Y,2DY,2

+F ′′(K2)

F ′(K2)K ′2

]+UC,1

UCM,2

UC,2M ′

2

=

{UCCUC

= − θC

and C2 = Y2

}=

=UC,1

[(1− θ)

D′Y,2DY,2

+D′KDK

+UCM,2

UC,2M ′

2

]+UC,1

[F ′′(K2)

F ′(K2)− θ F

′(K2)

F (K2)

]K ′2.

Equating these derivatives and rearranging delivers

K ′2 =

θC1Y1

D′Y,1DY,1

+ (1− θ)D′Y,2

DY,2+(

1 + θC1

K2

DK

)D′KDK

θC1

1DK

+ θ F′(K2)

F (K2)− F ′′(K2)

F ′(K2)

+−UCM,1

UC,1M ′

1 +UCM,2UC,2

M ′2

θC1

1DK

+ θ F′(K2)

F (K2)− F ′′(K2)

F ′(K2)

(3.71)


The following proposition can now be stated:

Proposition 3.3. In the coal case, the �rst-period fossil-fuel use is ex-ogenously determined. The second-period fossil-fuel use depends on DY,1,DY,2, DK, M1 and M2 as follows

∂B2

∂DY,1

≥ 0,∂B2

∂DY,2

≤ 0,∂B2

∂DK

≥ 0,∂B2

∂M1

≤ 0 and∂B2

∂M2

≥ 0

Proof. First-period capital is given. Since this means that G1 is given,equation (3.68) implies that �rst-period fossil-fuel use is given. Forsecond-period fossil-fuel use, (3.68) gives that dB2

dK2> 0. From (3.70)

it follows that F ′ > 0 and that F ′′ < 0, since they have the same signsas ∂Y

∂Kand ∂2Y

∂K2 , respectively. The results then follow from equation(3.71) since the denominator on the right-hand side is positive, θ ≥ 1and UCM ≥ 0.

The results can be interpreted as follows. First-period capital, andtherefore also fossil-fuel use, is given. If DY,1 decreases, production inthe �rst period and the amount of resources that can be divided between�rst-period consumption and investment into second-period capital bothdecrease. This leads to decreases in both C1 and K2. The decrease in K2

implies a decrease in B2. If, instead, DY,2 decreases, then the marginalproduct of second-period capital must decrease while the marginal util-ity of second-period consumption increases. Which of these e�ects dom-inates depends on the sign of θ − 1. If θ > 1, the e�ect on marginalutility from consumption in the second period dominates. First-periodinvestment then increases giving increased G2 and B2.

If DK decreases, there is a direct e�ect of a decrease in second-periodcapital. There is also a decrease in the amount of second-period cap-ital that each unit of savings gives. Counteracting this, the value ofsecond-period capital increases both due to increased marginal productof capital and the increased marginal utility from consumption. So theremay be an increase in �rst-period investment, but this e�ect will not belarge enough to counteract the e�ects of increased depreciation. Thus, adecrease inDK leads to a decrease in second-period capital which impliesa decrease in second-period fossil-fuel use.

Changes in M1 and M2 do not a�ect the production possibilities.Instead, they a�ect the relative value of consumption between the twoperiods, thus inducing a redistribution of consumption between the twoperiods. This occurs by changing �rst-period investment. If the value of�rst-period consumption decreases relative to second-period consump-tion, �rst-period investment increases. This increases the amount of


second-period capital which implies an increase in second-period fossil-fuel use. If the value of �rst-period consumption increases relative tosecond-period consumption, the opposite will occur.

3.3.3 Ampli�cation or dampening of climate change

Sections 3.3.1 and 3.3.2 derived the results of how exogenous changesa�ected the equilibrium allocation. The intended interpretation is thatthese changes are in fact driven by climate change which is driven byfossil-fuel use and, therefore, endogenous. In this section, I will discusshow, in a model with endogenous climate, the change in the allocationdue to the e�ects of climate change would a�ect fossil-fuel use. In par-ticular I will discuss whether the reactions to the realization that thereis climate change might amplify or dampen climate change.

To begin with, how will climate change manifest itself in the model?Climate change will decrease DY and DK and increase M . Since DK

gives the amount of depreciation between the periods, the e�ects of cli-mate change can be interpreted as a decrease in DK . For changes in DY

and M I have to determine if they primarily a�ect the period 1 or pe-riod 2 values. As has been seen above, the timing of the changes matterscrucially. Thus, for changes in DY and M I must determine what thee�ects will be in the di�erent periods. On the one hand, climate changeis a slow process; one the other hand, the use of a two period modelimplies that a time period should be considered as being relatively long.For the purpose of the present discussion, I will make the assumptionthat the e�ects on the second-period values are larger than the e�ectson the �rst-period values.

To say whether the reactions to climate change will dampen or am-plify the change, I must also determine what I mean by ampli�cationor dampening. These interpretations will be somewhat di�erent in theoil and coal cases. In the oil case, total fossil-fuel use is exogenouslygiven by the total initial supply. Thus, only its timing is endogenouslydetermined. I will then interpret a reallocation of fossil-fuel use fromthe second to the �rst period as an ampli�cation of climate change. Inthe coal case, �rst-period fossil-fuel use is determined by �rst-periodcapital. Second-period fossil-fuel use is endogenously determined by the�rst-period choice of second-period capital. In that case, I will interpretan increase in second-period fossil-fuel use as an ampli�cation of climatechange.

Based on the assumption about what the e�ects of climate changewill be, and the interpretation of what ampli�cation of climate changemeans, propositions 3.2 and 3.3 give the e�ects in the di�erent cases.Proposition 3.2 says that in the oil case, with exogenously given alterna-

3.4. DISCUSSION 133

tive energy, climate e�ects on productivity or utility will dampen climatechange while climate e�ects on depreciation will amplify climate change.Proposition 3.3 says that in the coal case, climate e�ects on productivityor utility will amplify climate change while climate e�ects on deprecia-tion will dampen climate change. Thus, we conclude that climate e�ectson depreciation will have the opposite e�ect compared to climate e�ectsa�ecting productivity or utility directly.

Furthermore, these e�ects will be reversed in the coal case comparedto the oil case: the dampening or amplifying e�ects of climate change,for a given type of damages, are the opposite in the oil and coal case.3

Still, the e�ects are similar in both cases in terms of in which directionthe intertemporal change goes. The di�erence is largely driven by whatis exogenously given. In the oil case total fossil-fuel use is given while inthe coal case, �rst-period fossil-fuel use is given.

3.4 Discussion

This chapter has investigated how the nature of climate externalitiesin�uences our analysis of the e�ects of climate change on the economy.I looked at damages to productivity, capital depreciation and utilitydirectly and I carried out the analysis in two complementary ways.

Firstly, I derived a formula that, under some speci�c assumptions,gave the optimal per unit fossil-fuel tax. The formula is very simple andstates that the optimal tax in any period, in relation to total productionin that period, equals a speci�c constant. This constant is a sum ofthree di�erent parts, one for each type of climate e�ect. This formulaserves as an aggregating device: it allows us to combine di�erent typesof e�ects of climate change into one measure.

The formula is very appealing in its simplicity. Its derivation requiredspeci�c assumptions for some aspects of the model, while other aspectsremained very general. Perhaps the strongest assumptions are thoseregarding the shape of the damage functions. While these assumptionsmay be reasonable for moderate climate change, they cannot capturethe risks of catastrophic climate change.

Secondly,using a two-period model, I considered how the di�erentpossible e�ects of climate change a�ect fossil-fuel use. I considered twodi�erent cases. In the oil case, fossil fuel is costlessly extracted from agiven total supply. In that case, all fossil fuel is always used. In the coalcase, the extraction of fossil fuel requires inputs, but the scarcity of thefossil fuel does not matter. For the oil case, I considered one case wherealternative energy was exogenously given, and one case where alternative

3At least if considering the oil case with exogenously given alternative energy


energy was endogenously determined by the amounts of inputs used inthe generation of it. The main conclusion from this analysis is that thequalitative results depend crucially on the assumptions made. Thus, theassumptions made need to be carefully motivated.

In the two-period model, I treated climate as exogenous. I do notconsider this assumption very problematic. As discussed, the endoge-nous responses may either dampen or amplify climate change, but itdoes not seem likely that they would overturn the qualitative resultsregarding the signs of the changes in fossil-fuel use.

Restricting the analysis to a two-period model means that it is onlypossible to have changes between the �rst and the second periods. In amulti-period model, more complicated patterns of changes are possible.My conjecture is that the qualitative results derived here would gener-alize to a multi-period model where the climate changes monotonically.Since the climate is expected to deteriorate for quite a long time, therestriction to a two-period model need therefore not be so restrictive.

References

Acemoglu, D., P. Aghion, L. Bursztyn & D. Hemous, 2012, �The En-vironment and Directed Technical Change�, The American EconomicReview, 102(1): 131-66.

Dasgupta, P. & G. Heal, 1974, �The Optimal Depletion of ExhaustibleResources�, Review of Economic Studies, 41, 3-28.


Greenwood. J., Z. Hercowitz & G. W. Hu�man, 1988, �Investment,Capacity Utilization, and the Real Business Cycle�, The American Eco-nomic Review, Vol. 78, No. 3 (June), 402-417.

Hassler, J., P. Krusell & C. Olovsson, 2011, �Energy-Saving TechnicalChange�, mimeo , Institute for International Economic Studies, Stock-holm University

Hotelling, H., 1931, "The Economics of Exhaustible Resources", TheJournal of Political Economy, 39:2, 137-175

IPCC, 2007, �Contribution of Working Group II to the Fourth Assess-ment Report of the Intergovernmental Panel on Climate Change�, M. L.Parry, O. F. Canziani, J. P. Palutikof, P. J. van der Linden & C.E. Han-son (eds), Cambridge University Press, Cambridge, United Kingdom.

Layard, R., S. Nickell & G. Mayraz, 2008, �The marginal utility of in-come� Journal of Public Economics 92:1846-1857.

Nordhaus, W. and J. Boyer, 2000, Warming the World: Economic Mod-els of Global Warming, MIT Press, Cambridge, MA..

Sterner, T. & U. M. Persson, 2008, �An even sterner review: Introducingrelative prices into the discounting debate.� Review of EnvironmentalEconomics and Policy, 2 (1): 61-76.

Stiglitz, J., 1974, �Growth with Exhaustible Natural Resources: E�cientand Optimal Growth Paths.�, The Review of Economic Studies, Vol. 41:123-137.

van der Ploeg, F. & C. Withagen, C. �Is there really a green paradox�,Forthcoming in Journal of Environmental Economics and Management

135


Weitzman, M. L., 2010, �What is the `Damages Function' for GlobalWarming - and What Di�erence Might it Make?�, Climate Change Eco-nomics, 1(1): 57-69.

3.A. DERIVATIVES OF A CES PRODUCTION FUNCTION 137

3.A Derivatives of a CES production function

The production function is

F (G,E) =[γGG

σ−1σ + γEE

σ−1σ

] σσ−1

.

This yields that

FG = γG

(F

G

) 1σ

FGG =FG

(1

σ

FGF− 1

σ

1

G

)= − 1

σ

FGGγE

(E

F

)σ−1σ

= − 1

σ

FGFEF

E

G

FE = γE

(F

E

) 1σ

FEE =FE

(1

σ

FEF− 1

σ

1

E

)= − 1

σ

FEEγG

(G

F

)σ−1σ

= − 1

σ

FGFEF

G

E

FEG =FGE =1

σ

FGFEF

.

The following combinations of derivatives are utilized in the derivations:

FGGFG− θFG

F=− 1

G

1σFEE + θFGG

F

θFEF− FEG

FG=

(θ − 1

σ

)FEF

FEGFE− FGG

FG=

1

σ

1

GFEGFG− FEE

FE=

1

σ

1

E

FEEFGG =F 2EG =

1

σ2

F 2GF

2E

F 2.

3.B Calculations for the oil case

3.B.1 Calculations for the oil case with endogenous

alternative energy

Calculation of GY,B and GS,B

When di�erentiating with respect to B, total G is �xed. This meansthat GY,B = −GS,B.


Di�erentiating both sides of (3.61) with respect to B delivers

dFY,GdB

=FY,G

[FY,GGFY,G

GY,B +FY,GEFY,G

(FE,B + FE,SASGS,B)

]=FY,G

[FY,GEFY,G

FE,B +

(FY,GEFY,G

FE,SAS −FY,GGFY,G

)GS,B

]= {(3.61)} = FY,G

[FY,GEFY,G

FE,B +

(FY,GEFY,E

− FY,GGFY,G

)GS,B

]=FY,G

[FY,GEFY,G

FE,B +1

σY

1

GY

GS,B

]

and

dFY,EFE,SASdB

=FY,EFE,SAS

[FY,GEFY,E

GY,B +FY,EEFY,E

(FE,B + FE,SASGS,B)

]+FY,EFE,SAS

[FE,BSFE,S

+FE,SSFE,S

ASGS,B

]= {(3.61)} =

=FY,G

[(FY,EEFY,E

FY,GFY,E

+FE,SSFE,S

AS −FY,GEFY,E

)GS,B

]+FY,G

[FY,EEFY,E

FE,B +FE,BSFE,S

]=FY,G

[((FY,EEFY,E

− FY,GEFY,E

)FY,GFY,E

+FE,SSFE,S

AS

)GS,B

]+FY,G

[FY,EEFY,E

FE,B +FE,BSFE,S

]=FY,G

[(FE,SSFE,S

AS −1

σY

1

E

FY,GFY,E

)GS,B

]+FY,G

[FY,EEFY,E

FE,B +FE,BSFE,S

].

3.B. CALCULATIONS FOR THE OIL CASE 139

Equating these derivatives and solving for GS,B produces

GS,B =−GY,B =

FY,EEFY,E

FE,B +FE,BSFE,S

− FY,EGFY,G

FE,B

1σY

1GY

+ 1σY

1E

FY,GFY,E− FE,SS

FE,SAS

=

(FY,EEFY,E

− FY,EGFY,G

)FE,B +

FE,BSFE,S

1σY

1GY

+ 1σY

1E

FY,GFY,E− FE,SS

FE,SAS

=− 1σY

1EFE,B + 1

σE

FE,BFE

1σY

1GY

+ 1σY

1E

FY,GFY,E− FE,SS

FE,SAS

=

(1σE− 1

σY

)1EFE,B

1σY

1GY

+ 1σY

1E

FY,GFY,E− FE,SS

FE,SAS

=

{−FE,SSFE,S

=1

σE

FE,BFE

B

SAS =

1

σE

FE,BB

FE

1

GS

}=

=

(1σE− 1

σY

)1EFE,B

1σY

FYFY,EE

1GY

+ 1σE

FE,BB

FE

1GS

. (3.72)

Calculation of GY,G and GS,G

When di�erentiating with respect to G, GY,G+GS,G = 1. Di�erentiatingboth sides of (3.61) with respect to G yields

dFY,GdG

=FY,G

[FY,GGFY,G

GY,G +FY,GEFY,G

FE,SASGS,G

]=FY,G

[FY,GGFY,G

+

(FY,GEFY,G

FE,SAS −FY,GGFY,G

)GS,G

]= {(3.61)} = FY,G

[FY,GGFY,G

+

(FY,GEFY,E

− FY,GGFY,G

)GS,G

]=FY,G

[FY,GGFY,G

+1

σY

1

GY

GS,G

]


and

dFY,EFE,SASdG

=FY,EFE,SAS

[FY,GEFY,E

GY,G +FY,EEFY,E

FE,SASGS,G

]+FY,EFE,SAS

FE,SSFE,S

ASGS,G = {(3.61)} =

=FY,G

(FY,EEFY,E

FE,SAS +FE,SSFE,S

AS −FY,GEFY,E

)GS,G

+FY,GFY,GEFY,E

= {(3.61)} =

=FY,G

[(FY,EEFY,E

− FY,GEFY,E

)FY,GFY,E

+FE,SSFE,S

AS

]GS,G

+FY,GFY,GEFY,E

=FY,G

[(− 1

σY

1

E

FY,GFY,E

+FE,SSFE,S

AS

)GS,G +

FY,GEFY,E

].

Equating these derivatives and solving for GS,G leads to

GS,G =

FY,GEFY,E

− FY,GGFY,G

1σY

1GY

+ 1σY

1E

FY,GFY,E− FE,SS

FE,SAS

=

{−FE,SSFE,S

=1

σE

FE,BFE

B

SAS =

1

σE

FE,BB

FE

1

GS

}=

=1σY

1GY

1σY

FYFY,EE

1GY

+ 1σE

FE,BB

FE

1GS

∈ (0, 1) (3.73)

and

GY,G = 1−GS,G =

1σY

1E

FY,GFY,E− FE,SS

FE,SAS

1σY

FYFY,EE

1GY

+ 1σE

FE,BB

FE

1GS

∈ (0, 1).

Calculation of derivatives of F (B,K)

Di�erentiating (3.62) delivers

FY,B =FY,GGY,B + FY,E (FE,B + FE,SASGS,B)

= (FY,G − FY,EFE,SAS)GY,B + FY,EFE,B

= {(3.61)} = FY,EFE,B

FY,K =FY,GGY,GGK + FY,EFE,SASGS,GGK

=FY,GGK + (FY,EFE,SAS − FY,G)GS,GGK

= {(3.61)} = FY,GGK .


Di�erentiating FY,B gives

FY,BB = (FY,EGGY,B + FY,EE (FE,B + FE,SASGS,B))FE,B

+FY,E (FE,BB + FE,BSASGS,B) = {(3.61)} =

=FY,EEF2E,B + FY,EFE,BB

+

(FY,G

(FY,EGFY,G

− FY,EEFY,E

)FE,B − FY,EFE,BSAS

)GY,B

=FY,EEF2E,B + FY,EFE,BB + FY,G

(1

σY− 1

σE

)FE,BFE

GY,B

FY,BK = (FY,EGGY,GGK + FY,EEFE,SASGS,GGK)FE,B

+FY,EFE,BSASGS,GGK = {(3.61)} =

=FY,EGGKFE,B

+

[(FY,EE

FY,GFY,E

− FY,EG)FE,B + FY,EFE,BSAS

]GS,GGK

=FY,EGGKFE,B

+FY,G

[(FY,EEFY,E

− FY,EGFY,G

)FE,B +

FE,BSFE,S

]GS,GGK

=FY,EGGKFE,B − FY,G(

1

σY− 1

σE

)FE,BE

GS,GGK .

Di�erentiating FY,K produces

FY,KK = [FY,GGGY,GGK + FY,EGFE,SASGS,GGK ]GK + FY,GGKK

=

[FY,GG (1−GS,G) + FY,EG

FY,GFY,E

GS,G

]G2K + FY,GGKK

=FY,GGG2K + FY,GGKK + FY,G

(FY,EGFY,E

− FY,GGFY,G

)GS,GG

2K

=FY,GGG2K + FY,GGKK +

1

σYFY,G

1

GY

GS,GG2K .


Substituting for GY,B from (3.72) in FY,BB and rewriting yields

FY,BB =FY,EEF2E,B + FY,EFE,BB

+

(1

σY− 1

σE

)FY,GFE,B

1

E

(1σY− 1

σE

)1EFE,B

1σY

FYFY,EE

1GY

+ 1σE

FE,BB

FE

1GS

=

{FY,GFY,EE

= − FY,G1σY

FY,GFY,EFY

GYE

= − FY1σYFY,EE

1

GY

E2

}=

=FY,EE

1−

(1σY− 1

σE

)2FY

FY,EE1GY

1σ2Y

FYFY,EE

1GY

+ 1σY σE

FE,BB

FE

1GS

F 2E,B + FY,EFE,BB < 0.

The parenthesis in the last expression lies between 0 and 1 since

0 <

(1

σY− 1

σE

)2FY

FY,EE

1

GY

<1

σ2Y

FYFY,EE

1

GY

.

Substituting GS,G from (3.73) into FY,BK implies

FY,BK =FY,EGGKFE,B

−FY,G(

1

σY− 1

σE

)FE,BE

1σY

1GY

1σY

FYFY,EE

1GY

+ 1σE

FE,BB

FE

1GS

GK

=

{FY,GFY,EG

=FY,G

1σY

FY,EFY,GFY

=

FYFY,E

1σY

}=

=FY,EG

1−

(1σY− 1

σE

)FY

FY,EE1GY

1σY

FYFY,EE

1GY

+ 1σE

FE,BB

FE

1GS

FE,BGK > 0.

The parenthesis in the last expression lies between 0 and 1 since

0 <1

σY

(1

σY− 1

σE

)FY

FY,EE

1

GY

<1

σ2Y

FYFY,EE

1

GY

.


Substituting for GS,G in FY,KK gives

FY,KK =FY,GGG2K + FY,GGKK

+1

σYFY,G

1

GY

1σY

1GY

1σY

FYFY,EE

1GY

+ 1σE

FE,BB

FE

1GS

=

{1σYFY,G

1GY

FY,GG=

1σYFY,G

1GY

− 1σY

FY,GFY,EFY

EGY

= − FYFY,EE

}=

=FY,GG

1−1σY

FYFY,EE

1GY

1σY

FYFY,EE

1GY

+ 1σE

FE,BB

FE

1GS

G2K + FY,GGKK < 0.

The parenthesis in the last expression lies between 0 and 1.

Calculating expressions (3.58) and (3.59)

Using the derivatives of F from the previous section, we obtain

FY,K

FY=FY,GFY

GK

FY,BK

FY,B=FY,EGGKFE,B − FY,G

(1σY− 1

σE

)FE,BEGS,GGK

FY,EFE,B

=FY,EGFY,E

GK −(

1

σY− 1

σE

)FY,GFY,EE

GS,GGK

FY,KK

FY,K=FY,GGG

2K + FY,GGKK + FY,G

1σY

1GYGS,GG

2K

FY,GGK

=FY,GGFY,G

GK +GKK

GK

+1

σY

1

GY

GS,GGK .

These can be combined so that we can express (3.58) as follows:

θFY,K

FY− FY,BK

FY,B= θ

FY,GFY

GK −FY,EGFY,E

GK +

(1

σY− 1

σE

)FY,GFY,EE

GS,GGK

=

[(θ − 1

σY

)FY,GFY

+

(1

σY− 1

σE

)FY,GFY,EE

GS,G

]GK

=

[θ − 1

σY+

(1

σY− 1

σE

)FY

FY,EEGS,G

]FY,GFY

GK .


GS,G from (3.73) can be rewritten as

GS,G =1σY

1GY

1σY

1GY

+ 1σY

1E

FY,GFY,E− FE,SS

FE,SAS

=1σY

1σY

+ 1σY

FY,GGYFY,EE

− FE,SSFE,S

ASGY

=1σY

1σY

FYFY,EE

+ 1σE

FE,BB

FESASGY

=1σY

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

.

Using this, we obtain

θFY,K

FY− FY,BK

FY,B=

θ − 1

σY+

1σY

(1σY− 1

σE

)FY

FY,EE

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

FY,GFY

GK

=

(θ − 1

σY

)(1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

)1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

FY,GFY

GK

+

1σY

(1σY− 1

σE

)FY

FY,EE

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

FY,GFY

GK

=

(θ − 1

σE

)1σY

FYFY,EE

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

FY,GFY

GK

+

(θ − 1

σY

)1σE

FE,BB

FE

GYGS

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

FY,GFY

GK .

Furthermore, the condition for the allocation of G between the sec-tors, equation (3.61), implies that

FYFY,EE

=FY,EE + FY,GGY

FY,EE= 1 +

FY,GGY

FY,EE= {(3.61)} = 1 + FE,SAS

GY

E

= {S = ASGS} = 1 +FE,SS

FE

GY

GS

.


Using this equation, we see that

θFY,K

FY− FY,BK

FY,B=

(θ − 1

σE

)1σY

(1 +

FE,SS

FE

GYGS

)1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

FY,GFY

GK

+

(θ − 1

σY

)1σE

FE,BB

FE

GYGS

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

FY,GFY

GK

=

{FE,BB

FE+FE,SS

FE= 1

}

=

(θ − 1

σE

)1σY

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

FY,GFY

GK

+

(θ(

1σY

FE,SS

FE+ 1

σE

FE,BB

FE

)− 1

σEσY

)GYGS

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

FY,GFY

GK .

Similarly, calculating the expression in (3.59) instead, we arrive at

FY,BK

FY,B− FY,KK

FY,K=

(FY,EGFY,E

− FY,GGFY,G

)GK −

GKK

GK

−[(

1

σY− 1

σE

)FY,GFY,EE

+1

σY

1

GY

]GS,GGK

=1

σY

1

GY

GK −GKK

GK

−[

1

σY

FYFY,EE

− 1

σE

FY,GGY

FY,EE

]1

GY

GS,GGK .

Using GK = αGK, GKK = (α− 1)GK

K, this equation becomes

FY,BK

FY,B− FY,KK

FY,K=

1

σYαG

GY

1

K− (α− 1)

1

K

−[

1

σY

FYFY,EE

− 1

σE

FY,GGY

FY,EE

]GS,Gα

G

GY

1

K

=1

K+ α

[1

σY

G

GY

− 1

]1

K

−α[

1

σY

FYFY,EE

− 1

σE

FY,GGY

FY,EE

]GS,G

G

GY

1

K.


Substituting for GS,G delivers

FY,BK

FY,B− FY,KK

FY,K=

1

K+ α

[1

σY

G

GY

− 1

]1

K

−α1σY

[1σY

FYFY,EE

− 1σE

FY,GGYFY,EE

]1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

G

GY

1

K

=

{Canceling two terms

1

σ2Y

FYFY,EE

}=

1

K− α 1

K

+α1

σY σE

FE,BB

FE

GYGS

+FY,GGYFY,EE

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

G

GY

1

K.

Using equation (3.61), we obtain

FY,GGY

FY,EE=FE,SS

FE

GY

GS

⇒ FE,BB

FE

GY

GS

+FY,GGY

FY,EE=GY

GS

.

Furthermore,

FYFY,EE

= 1 +FE,SS

FE

GY

GS

⇒ 1

σY

FYFY,EE

+1

σE

FE,BB

FE

GY

GS

=1

σY+

(1

σY

FE,SS

FE+

1

σE

FE,BB

FE

)GY

GS

implies

FY,BK

FY,B− FY,KK

FY,K=α

1σY σE

GYGS

GGY−(

1σY

+(

1σY

FE,SS

FE+ 1

σE

FE,BB

FE

)GYGS

)1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

1

K

+1

K

=

{GY

GS

G

GY

=G

GS

= 1 +GY

GS

}

=α

1σY

1−σEσE

+(

1σY

1−σEσE

FE,SS

FE+ 1

σE

1−σYσY

FE,BB

FE

)GYGS

1σY

FYFY,EE

+ 1σE

FE,BB

FE

GYGS

1

K+

1

K.


3.B.2 Determining the sign of ξB

From the derivation of (3.49), it can be seen that ξB is given by

ξB =θ

C1

Y1FY,B,1FY,1

(FY,BK,2FY,B,2

− FY,KK,2FY,K,2

)− 1

DK

(FY,BB,1FY,B,1

+FY,BB,2FY,B,2

− FY,BK,2FY,K,2

)FY,BK,2FY,B,2

− FY,KK,2FY,K,2

+FY,BB,1

FY,B,1

FY,KK,2FY,K,2

− θ FY,K,2FY,2

FY,BK,2FY,B,2

− FY,KK,2FY,K,2

+

(θFY,K,2FY,2

− FY,KK,2FY,K,2

)(FY,BK,2FY,K,2

− FY,BB,2FY,B,2

)FY,BK,2FY,B,2

− FY,KK,2FY,K,2

+

(FY,BK,2FY,B,2

− FY,KK,2FY,K,2

)(θFY,B,2FY,2

− FY,BK,2FY,K,2

)FY,BK,2FY,B,2

− FY,KK,2FY,K,2

(3.74)

Exogenous alternative energy

Using the derivatives of F with exogenous alternative energy from sec-tion 3.3.1 (in the subsection starting on page 120), one can observe thatthe �rst three terms in the expression (3.74) for ξB are positive. In thefourth term, the expression

−

(FY,BK,2FY,B,2

− FY,KK,2FY,K,2

)FY,BK,2FY,K,2

FY,BK,2FY,B,2

− FY,KK,2FY,K,2

=

FY,KK,2FY,K,2

FY,BK,2FY,K,2

− F 2Y,BK,2

FY,B,2FY,K,2

FY,BK,2FY,B,2

− FY,KK,2FY,K,2

is negative. The �rst part of this expression can be canceled by anidentical term with opposite sign from the third term of (3.74). In orderto cancel the second part, it can �rst be noted that using the derivativesin appendix 3.A, F 2

Y,EG = FY,EEFY,GG. Then we obtain

FY,BBFY,KK −(FY,BK

)2

=FY,EE(FE,B)2FY,GG(GK)2

+FY,EE(FE,B)2FY,GGKK

+FY,EFE,BBFY,GG(GK)2

+FY,EFE,BBFY,GGKK

− (FY,EG)2 (FE,B)2(GK)2

=FY,EE(FE,B)2FY,GGKK

+FY,EFE,BBFY,GG(GK)2

+FY,EFE,BBFY,GGKK > 0.


Using this fact, and a term from the third term of (3.74), we deducethat

FY,KK,2FY,K,2

FY,BB,2FY,B,2

− F 2Y,BK,2

FY,B,2FY,K,2

FY,BK,2FY,B,2

− FY,KK,2FY,K,2

=

FY,KK,2FY,BB,2−F 2Y,BK,2

FY,B,2FY,K,2

FY,BK,2FY,B,2

− FY,KK,2FY,K,2

> 0.

These arguments can be summarized to conclude

ξB > 0

Endogenous alternative energy

In section 3.B.1 (in the part starting on page 140) it is possible to observethat also with endogenous alternative energy we have

FY,BB < 0, FY,BK > 0 and FY,KK < 0.

This means that the �rst two terms in (3.74) are positive. The last twoterms in (3.74) are(

θFY,K,2FY,2

− FY,KK,2FY,K,2

)(FY,BK,2FY,K,2

− FY,BB,2FY,B,2

)FY,BK,2FY,B,2

− FY,KK,2FY,K,2

+

(FY,BK,2FY,B,2

− FY,KK,2FY,K,2

)(θFY,B,2FY,2

− FY,BK,2FY,K,2

)FY,BK,2FY,B,2

− FY,KK,2FY,K,2

=2θ

FY,BKFY

+FY,BBFY,KKFY,BFY,K

− F 2Y,BK

FY,BFY,K− θ FY,K FY,BB

FY FY,B− θ FY,BFY,KK

FY FY,K

FY,BK,2FY,B,2

− FY,KK,2FY,K,2

.

All the terms in the denominator of this expression are positive except

for − F 2Y,BK

FY,BFY,K. It can, however, be shown that

2θFY,BK

FY+FY,BBFY,KK

FY,BFY,K−

F 2Y,BK

FY,BFY,K> 0.

This implies that ξB > 0 also in this case.

Chapter 4

Indirect E�ects of Climate

Change

4.1 Introduction

The direct e�ects of climate change are expected to di�er a great dealacross the world. Some countries are expected to su�er very severeconsequences, where perhaps the most extreme cases are countries thatare under serious risk of �ooding due to sea-level rise. In contrast, somecountries will only face small negative consequences of climate change,and could even bene�t from moderate climate change. Apart from thedi�erences in physical e�ects, the capacity to adapt to the changingconditions implied by climate change also di�ers a great deal acrosscountries.

The contribution to climate change, through the emission of green-house gases, also varies signi�cantly across countries. As an example,Bangladesh, a country that would be hit very hard by the e�ects ofsea-level rise, contributes about 0.3% of world emissions, while the UScontributes about 15-20% of world emissions (www.cait.org).

So both the consequences of, and the contribution to, climate changevary very much across countries. A casual look at the vulnerability toclimate change and the current emissions of greenhouse gases suggestsan inverse relationship between these. This poses a problem for the mit-igation of climate change (reductions of emissions of greenhouse gases)since the countries that can contribute the most to mitigation e�ortstend to be the countries least interested in avoiding climate change.

Measurements of vulnerability to climate change are typically basedon the direct e�ects that will occur in a country. However, the assessmentof the consequences of climate change for a country should really bebased on total general equilibrium e�ects.

149

150 CHAPTER 4. INDIRECT EFFECTS OF CLIMATE CHANGE

I will here consider two di�erent types of indirect e�ects that willcontribute to the total e�ects. First, I will consider trade in goods.Second, I will consider trade in �nancial instruments that can be usedto insure against weather induced shocks.

So the �rst channel of indirect e�ects that I consider is trade ingoods. A common way of modeling the damages caused by climatechange is to assume that climate change a�ects productivity. On theworld market, changes in productivity in a country will result in generalequilibrium e�ects on the world market prices of goods. In order tocapture the trade between countries that will be a�ected di�erently byclimate change, I will focus on trade in di�erent goods between countrieswith di�erent comparative advantages (rather than on trade in similargoods among countries with similar comparative advantages). Therefore,I model trade using a model of the Ricardian type.

The second channel of indirect e�ects that I consider is trade in �nan-cial instruments that can be used to insure against variability. Climatechange is expected to result in more variability in weather outcomes. Itis expected to lead to an increased probability, and severity, of extremeweather events such as droughts, �oods, cyclones and heatwaves (IPCC,2007). Financial markets will allow for some insurance against thesekinds of events. The prices of insuring instruments will depend on theworld distribution of weather shocks. To model the trade in �nancial in-struments, I set up a two-period endowment model. The second-periodendowments are stochastic and I assume there to be complete marketsto insure against this uncertainty. In this model, climate change acts bychanging the second-period endowment distribution.

The general principle for the indirect e�ects, for both channels, isthat a country bene�ts from changes that increase the world marketprice of goods or instruments of which the country is a net seller orchanges that decrease the world market price of goods or instruments ofwhich the country is a net buyer, and the other way around.

So, in general, the signs of the indirect e�ects will critically dependon the assumptions made about how the e�ects of climate change relateto the patterns of trade in goods and �nancial instruments.

I also apply these more general results to a two-country example.This example is intended to capture the asymmetry that can be seen be-tween rich and poor countries. Making the asymmetry stark, I assumethat emissions of greenhouse gases are only made in the rich country,while the e�ects of climate change are only felt in the poor country.Therefore, in this example, it would be di�cult to reach agreementsabout climate change mitigation since the mitigation e�orts must bemade by the rich country that is not, directly, a�ected by the negative


consequences of climate change. Under reasonable assumptions, I �ndthat the channel through trade in goods will have negative indirect ef-fects in the rich country, while the channel through trade in �nancialinstruments will have positive indirect e�ects in the rich country. Sothe channel through trade in goods will tend to make the interests ofthe countries more aligned with each other, while the channel throughtrade in �nancial instruments will make the interests of the countriesless aligned.

In the two-country case of trade in goods, I also consider how tari�sa�ect the incentives of the rich country to reduce emissions. In that case,I �nd that it may be possible for the poor country to induce the richcountry to reduce the emissions by threatening to increase the tari�s.

Looking at the previous literature, the most widely used models forstudying the interaction between the economy and the climate are theDICE and RICE models developed by Nordhaus (see e.g. Nordhaus andBoyer, 2000). These models have a homogeneous consumption good anddo not have any uncertainty so the mechanisms considered here are notpresent at all.

Regarding climate change and trade in goods, there is a relativelylarge literature on the pollution haven hypothesis, in general, and therisk of carbon leakage. This refers to the risk that unilateral e�orts toreduce emissions of greenhouse gases will just cause the emissions tomove to other countries. Overviews of this literature are provided byCopeland and Taylor (2004) and Antweiler et al. (2001). There is alsomore recent work that considers how endogenous technological changea�ects the risk of carbon leakage (see, e.g., Di Maria and Smulders 2004;Di Maria and van der Werf, 2008; Golombek and Hoel, 2004; Hemous2012). I will not have any carbon leakage e�ects in the models I usehere.

Regarding insurance against climate related events, Arrow et al.(1996) and Chichilnisky (1998) both discuss some problems related tothe ability of �nancial markets to provide insurance against climate re-lated events. I will say more about this in section 4.4. Arrow et al.(1996) note that if there were markets for insurance against climate re-lated events, the insurance premia would inequitably be borne by thoseexposed to the risks. This point is related to the way that trade in �-nancial instruments makes the interests of the countries less aligned inthis chapter.

The rest of this chapter consists of three sections. Sections 4.2 and4.3 set up and analyze models with trade in goods and insuring �nancialinstruments, respectively. Each of these sections starts with setting up amodel with many countries and deriving results about possible changes.


These general results are then applied to the two-country case. For tradein goods, the two-country case also has a section with tari�s. Section4.4 provides conclusions, a discussion of the results and suggestions forfuture work.

4.2 Trade in goods

In this section, I will set up a model with trade in goods between coun-tries and use this model to analyze the total welfare e�ects of changesin productivities. If the productivity changes in one country, e.g., dueto climate change, this will, in addition to having a direct e�ect on theincome in that country, a�ect the world market prices of goods. Thesechanges in world market prices a�ect the welfare in other countries.

A signi�cant share of GDP consists of goods that are traded inter-nationally. Part of this trade is trade in similar goods among similarcountries. Another part of this trade is trade in di�erent goods betweencountries with di�erent comparative advantages. The purpose of thischapter is to capture trade between countries that are a�ected di�er-ently by climate change and, in the two-country case, it is intended tocapture the trade between the north and the south. Therefore, the tradein di�erent goods between countries that di�er in their comparative ad-vantages seems to be the most relevant case. The trade model used herewill therefore be of the Ricardian type where countries choose what toproduce based on their comparative advantages. The comparative ad-vantages are determined by the countries' good-speci�c productivities.In the model that I set up here, I will assume that the comparativeadvantages are strong enough to induce each country to specialize inthe production of one good and I will not explicitly model the choiceof what to produce. In a more general model, the choice of what toproduce would be endogenous based on the good-speci�c productivities.Each country could also produce many goods. The e�ects of changesin the good-speci�c productivities of a county, rather than the generalproductivity, could then be analyzed. Assuming that each country isspecialized in the production of one good simpli�es the analysis, butthe main conclusions from the model generalize relatively straightfor-wardly to the more general setting, at least as long as the changes inproductivities are such that the signs of net exports do not change.

I will �rst, in section 4.2.1, set up a many country model and, insection 4.2.2, I will solve for the equilibrium allocation. In section 4.2.3,I will then consider the welfare e�ects of changes in productivities. Insection 4.2.4 I will look at the implications for the two-country (northand south) case. In section 4.2.5 I will consider how trading costs, in theform of import tari�s, will a�ect the incentives for the north to reduce

4.2. TRADE IN GOODS 153

the emissions of greenhouse gases. Finally, in section 4.2.6, I will brie�ydiscuss how the indirect e�ects would a�ect the distribution of costsassociated with mitigation policies.

4.2.1 Model setup

There are I di�erent countries and J di�erent goods with I ≥ J . Thereis a measure li of households in country i, each supplying one unit oflabor. Production is linear in the amount of labor used. As explainedabove, each country, i, specializes in producing one good, ji. Let

Ij = {i ∈ {1, . . . , I} : ji = j}

denote the set of countries that specialize in the production of good jand let ai be the productivity per unit of labor in country i. Assumealso that at least one country specializes in the production of each good,that is, Ij is non-empty for all j. This implies that total production ofgood j is

Aj =∑i∈Ij

aili. (4.1)

Let ci,j be the amount of good j that is consumed by the represen-tative household in country i. Preferences over consumption baskets(ci,1, . . . , ci,J) are given by

U(ci,1, . . . , ci,J) = u(ci)

where

ci =

(J∑j=1

(ci,j)σ−1σ

) σσ−1

and u(c) =c1−θ − 1

1− θ. (4.2)

Here, σ ≥ 0 measures the substitutability between the di�erent con-sumption goods and θ measures the curvature of the function givingutility from consumption of the aggregate consumption good ci. Giventhat a good here should rather be interpreted as a relatively wide cat-egory of goods, it seems reasonable to assume that the substitutabilitybetween di�erent goods is relatively low, that is, σ should be relativelylow.

Let pj be the price of good j. Markets, in each country, are assumedto be competitive and, initially, there are no trading costs. The pricetaking behavior that is implicit in the assumption of competitive marketsis very important here since the indirect e�ects go through changingprices.

Competitive markets also imply that the wage is equal to the marginalvalue of labor, that is, the wage in country i is wi = pjiai. So, the budget


constraint of the representative household in country i is given by

aipji =J∑j=1

pjci,j. (4.3)

The optimal consumption choice problem of the household is

maxci,1,...,ci,J

U(ci,1, . . . , ci,J) s.t. aipji =J∑j=1

pjci,j. (4.4)

The equilibrium allocation and prices must be such that the repre-sentative household in each country i maximizes utility, given prices, andsuch that the market clearing condition

Aj =I∑i=1

lici,j (4.5)

holds for each good j, where Aj is de�ned in (4.1).

4.2.2 Equilibrium allocation

To solve for the equilibrium allocation and prices, I will start by solv-ing the consumption choice problem of the representative household incountry i and imposing market clearing. As long as the consumptiongoods are not perfect substitutes, σ <∞, the household will choose aninterior consumption basket such that ci,j > 0 for all j. Therefore, I willnot include non-negativity constraints on the quantities consumed.

The Lagrangian of the household's utility maximization problem(4.4) is

L = U(ci,1, . . . , ci,J) + λi

[aipji −

J∑j=1

pjci,j

]where λi is a multiplier. The �rst-order condition with respect to ci,j is

λipj =∂

∂ci,ju(ci) = u′(ci)

∂ci∂ci,j

= u′(ci)

(cici,j

) 1σ

.

Comparing goods j1 and j2, we obtain

pj1pj2

=

(ci,j1ci,j2

)− 1σ

or ci,j1 =

(pj1pj2

)−σci,j2 . (4.6)

So the relative consumption of good j1 and j2 depends on the relativeprice and the substitutability. The relative consumption levels will bethe same for all countries.


Substituting (4.6) in the market clearing condition (4.5) gives thatfor each good j

Aj =I∑i=1

lici,j =I∑i=1

li

(pjpj

)−σci,j =

(pjpj

)−σ I∑i=1

lici,j

for an arbitrary j. The last sum is independent of j, implying that forany goods j1 and j2

pj1pj2

=

(Aj1Aj2

)− 1σ

. (4.7)

That is, the relative price depends on the relative supply and on thesubstitutability.

Using (4.6) in the budget constraint (4.3), for each i we arrive at

wi = pjiai =J∑j=1

pjci,j =J∑j=1

pj

(pjpj

)−σci,j = ci,jp

σj

J∑j=1

p1−σj

for any j. Comparing representative households in country i1 and i2, forany j, it holds that

ci1,jci2,j

=wi1wi2

.

Using the market clearing condition (4.5), for any j

Aj =I∑i=1

lici,j =I∑i=1

lici,jci,jci,j

= ci,j

I∑i=1

liwiwi

= ci,j

∑Ii=1 liwiwi

.

This implies that for any (i, j)

ci,j = Ajwi∑I

i′=1 li′wi′.

De�nesi =

wi∑Ii′=1 li′wi′

. (4.8)

This is the wealth share of a representative household in country i outof total world wealth. For any (i, j) we now have

ci,j = siAj.

Therefore, for any good j, the representative household in country iconsumes its wealth share of total production of that good. Furthermore,for any i

ci =

(J∑j=1

cσ−1σ

i,j

) σσ−1

=

(J∑j=1

(siAj)σ−1σ

) σσ−1

= si

(J∑j=1

Aσ−1σ

j

) σσ−1

.


Let

A =

(J∑j=1

Aσ−1σ

j

) σσ−1

(4.9)

be a measure of total world production, taking substitutability into ac-count. Note the similarity between the way that the di�erent Aj arecombined into A and the way that the ci,j are combined into ci in (4.2).We thus have

ci = siA (4.10)

for all i.Thus, we see that the equilibrium allocation is such that each house-

hold consumes its wealth share of the world production of each goodand, therefore, also of total world production.

What remains now is to express the wealth share in terms of exoge-nous objects. In order to do this, the prices implicit in the wages mustbe substituted for. Using that wi = aipji in (4.8) and using the relativeprice (4.7), the wealth share can be rewritten as

si =wi∑I

i′=1 li′wi′=

pjiai∑Ii′=1 li′pji′ai′

=pjiai∑J

j′=1 pj′∑

i′∈Ij′li′ai′

=pjiai∑J

j′=1 pj′Aj′=

(AjiAj

)− 1σpjai∑J

j′=1

(Aj′

Aj

)− 1σpjAj′

=A− 1σ

jiai∑J

j′=1 Aσ−1σ

j′

.

This yields

si =A− 1σ

jiai∑J

j′=1 Aσ−1σ

j′

. (4.11)

Since the relative prices and the allocation of consumption are nowcompletely determined in terms of the parameters and exogenously givenvariables, this completes the characterization of the equilibrium. Therepresentative household in country i consumes share si, given by (4.11),of the total production of each good j. The relative prices are given by(4.7). One degree of freedom remains in the price determination. This isbecause it is only the relative prices, and not the price level, that matter.The prices can be completely determined by, for example, choosing onegood as the numeraire or by de�ning a price index that is normalized.

4.2.3 Welfare e�ects of changes in the productivities

We can now turn to looking at the welfare e�ects of changes in pro-ductivities. The implied interpretation of these changes is that they are


the result of climate change. In section 4.2.6, I will brie�y discuss theimplications of the derived results have for the division of the costs ofclimate change mitigation.

Since the utility of the representative household in country i is strictlyincreasing in ci, utility will change in the same direction as ci. Startingfrom (4.10), the e�ect of a change in productivity ai on ci is

dcidai

= sidA

dai+dsidai

A.

The �rst e�ect can be seen as a size-of-the-pie e�ect and the seconde�ect as a share-of-the-pie e�ect.

Di�erentiating (4.9), we obtain

dA

dai=

σ

σ − 1

(J∑j=1

Aσ−1σ

j

) 1σ−1

σ − 1

σA− 1σ

jili = A

A− 1σ

jili∑J

j=1Aσ−1σ

j

> 0.

So, as expected, the size e�ect is positive for an increase in productivityin any country.

Di�erentiating (4.11) delivers

dsidai

= si

ddaiA− 1σ

jiai

A− 1σ

jiai−

ddai

∑Jj=1A

σ−1σ

j∑Jj=1 A

σ−1σ

j

.

From (4.1), the �rst derivative is

d

daiA− 1σ

jiai =

0 if ji 6= ji

− 1σA− 1σ−1

jiliai if ji = ji but i 6= i

A− 1σ

ji

(1− 1

σliaiAji

)if i = i

and the second derivative is

d

dai

J∑j=1

Aσ−1σ

j =σ − 1

σA− 1σ

jili.

Combining these derivatives gives the total change in the wealth shareof the representative household in country i

dsidai

=

si1−σσA− 1σ

jili∑J

j=1 Aσ−1σ

j

if ji 6= ji

− si liAji

(1 + 1

σ

(1−

Aσ−1σ

ji∑Jj=1 A

σ−1σ

ji

))if ji = ji but i 6= i

siliAji

(Ajiliai− 1 + σ−1

σ

(1−

Aσ−1σ

ji∑Jj=1 A

σ−1σ

j

))if i = i


Intuitively, the change in the wealth share can be divided into twodi�erent e�ects. The �rst e�ect is that a country's share of the total sup-ply of goods changes and the second is that the value (the relative price)of the particular good a country produces changes. If ai increases, theshare of goods that country i produces increases if i = i and decreasesotherwise. The relative value of the good that country i produces de-creases if ji = ji and increases if ji 6= ji.

In the �rst case above (ji 6= ji), an increase in ai decreases the totalshare of goods that a country produces but increases the relative price ofthe goods that the country produces. Which of the e�ects that dominatesdepends on the substitutability. For a low elasticity of substitution (σ <1), the price e�ect dominates and the wealth share of the representativehousehold in country i increases. For the second case (ji = ji but i 6= i),both e�ects go in the same direction and an increase in ai decreases thewealth share of the representative household in country i. For the casei = i, an increase in ai increases the total share of goods that the countryproduces but decreases the relative price of the goods that they produce,ji. If country i is the only country producing good ji, then Aji = liaiand

dsidai

=siliAji

σ − 1

σ

1−A

σ−1σ

ji∑Jj=1A

σ−1σ

j

an expression that has the same sign as σ−1

σ. So this derivative has the

opposite sign as compared to the case where ji 6= ji. If country i is notthe only country producing good ji, then Aji > liai and the derivativeis strictly positive for σ = 1 (and for any σ ≥ 1). This re�ects that theprice e�ect of the productivity in country i is weakened if other countriesproduce the same good and then more complementarity is needed for theprice e�ect to dominate.

Putting together the size and share e�ects gives that

dcidai

=

1σsiAliAji

Aσ−1σ

ji∑Jj=1 A

σ−1σ

j

if ji 6= ji

−1+σσ

siAliAji

(1−

Aσ−1σ

ji∑Jj=1 A

σ−1σ

j

)if ji = ji but i 6= i

siAliAji

(Ajiliai− 1

σ

(1−

Aσ−1σ

ji∑Jj=1 A

σ−1σ

j

))if i = i

(4.12)

Given that welfare is strictly increasing in the consumption of theaggregate consumption good, the following proposition can be stated:


Proposition 4.1. Consider how a change in ai a�ects welfare in countryi. If i 6= i, then du(ci)

daiis negative if ji = ji and positive if ji 6= ji. If i = i

then

Sgn

(du(ci)

dai

)= Sgn

Ajiliai− 1

σ

1−A

σ−1σ

ji∑Jj=1A

σ−1σ

j

Proof. Since

Aσ−1σ

ji∑Jj=1 A

σ−1σ

j

< 1 and du(ci)dai

= u′(ci)dcidai

, this immediately fol-

lows from (4.12).

So the e�ect will be unambiguous if the productivity changes in othercountries. The sign in the case where the countries' own productivitychanges will depend on the parameters. In the expression for the deriva-

tive, Ajiliai≥ 1 while 1−

Aσ−1σ

ji∑Jj=1 A

σ−1σ

j

∈ (0, 1). This means that the derivative

is positive if σ ≥ liaiAji≤ 1 (implying that the derivative is always positive

if σ ≥ 1). The ratio liaiAji

gives the share of the total supply of good jithat is produced by country i. The smaller is this share, the weakeris the price e�ect of the productivity in country i and the stronger thecomplementarity between the goods must be for the price e�ect to domi-nate. To get some further sense of when the e�ect is positive or negative,consider the case where Aj = A for all j. Then, the outer parenthesisbecomes

A

liai− 1

σ

(1− 1

J

).

As before, this expression is always positive if liaiA≤ σ. If liai

A> σ, this

is (weakly) negative if and only if J ≥ 11−σ A

liai

.

So the conclusion is that an increase in productivity in country ihas a positive e�ect on welfare in country i 6= i if the countries arespecialized in producing di�erent goods, while it has a negative e�ectif the countries specialize in the production of the same type of goods.If the country's own productivity increases, the change in welfare dueto the change in relative prices will tend to decrease the welfare of therepresentative household in the country. Counteracting this e�ect is thee�ect of an increase in the amount of goods produced. The net e�ect isambiguous. So, in this setting it is perfectly possible that the derivativeis negative. That would mean that the representative household wouldbe better o� in a situation where productivity decreases. The reason isthat as the supplied quantity decreases, the relative price goes up. Whenthe interpretation is that productivity decreases due to climate change,


there would be likely to be other negative e�ects as well. There could bedecreased productivity in the production of non-traded goods, or therecould be other e�ects directly a�ecting welfare. So the possibility of anegative derivative should be interpreted with some caution. The e�ectson other countries should be much more robust.

4.2.4 The two-country case

Now, consider two countries. Country 1 can be considered as the de-veloped countries or the north, whereas country 2 can be considered asthe developing countries or the south. I will make the asymmetry be-tween the countries stark by assuming that only country 2 is a�ectedby climate change while all emissions of greenhouse gases are made bycountry 1. Under these assumptions, the question I will consider is howcountry 1 is indirectly a�ected by the e�ects in country 2. Furthermore,I will introduce trading costs in the form of import tari�s. The tari�swill then in�uence the welfare e�ects in country 1 of the climate inducedchanges in the productivity in country 2. Assuming that it is costly forcountry 1 to reduce the emissions of greenhouse gases, country 1 willthen trade o� the costs of emission reductions against the welfare e�ectsof climate change. This trade o� will depend on the tari�s and, there-fore, the e�ect of changing tari�s on the optimal amount of greenhousegas emissions in country 1 can be derived.

I will assume that there are two goods and that each country special-izes in producing one of the goods. In terms of the model set up above,this means that I = J = 2, Ij = {j} for all j and that ji = i for all i.

Assume now that climate change reduces productivity in country 2,a2. From equation (4.12), we obtain

dc1

da2

=1

σ

s1Al2A1

Aσ−1σ

2∑2j=1A

σ−1σ

j

> 0.

This implies that a decrease in productivity in country 2 decreaseswelfare in country 1. So, taking this channel into account should makecountry 1 want to reduce the emissions of greenhouse gases.

4.2.5 The two-country case with import tari�s

When trading goods, there are typically trading frictions. These can,among other things, be transportation costs, costs of adapting productsto new markets, costs of marketing products on new markets and tari�s.The kind of trading costs that I will choose to model is import tari�s.There are two reasons for why I choose to consider tari�s. Tari�s areclearly determined by policy, meaning that they could be used strategi-


cally to a�ect the outcomes. In particular, higher import tari�s in thesouth could potentially be used as a threat to induce the north to re-duce the emissions of greenhouse gases. Furthermore, tari�s do not leadto any production being destroyed. Therefore, the e�ects will only bedriven by the distortionary e�ects of trading costs and not by the lossof resources.

Model setup and equilibrium determination

Now assume that when households in country 1 buy good 2 a shareτ1 is subtracted and that when households in country 2 buy good 1 ashare τ2 is subtracted. Tari�s generate government revenues and I willassume that these are paid back lump sum to households. Let Ti be thelump-sum transfer to households in country i.

I normalize the price of good 1: p1 = 1.For given trading costs, the de�nition of an equilibrium is the same

as without trading costs. That is, the representative household in eachcountry maximizes the utility from consumption and all production isconsumed. I will not explicitly solve for prices and quantities here. In-stead, I will derive a set of equations that implicitly determines theequilibrium quantities. This will allow me to derive comparative staticsresults.

The optimization problem for the representative household in country1 is

maxc1,1,c1,2

U(c1,1, c1,2) s.t. a1 + T1 = c1,1 +p2

1− τ1

c1,2.

The �rst-order conditions with respect to c1,1 and c1,2 give

p2

1− τ1

=U2(c1,1, c1,2)

U1(c1,1, c1,2)=

(c1,2

c1,1

)− 1σ

⇒ p2 = (1− τ1)

(c1,2

c1,1

)− 1σ

. (4.13)

Similarly, the optimization problem for the representative household incountry 2 is

maxc2,1,c2,2

U(c2,1, c2,2) s.t. p2a2 + T2 =c2,1

1− τ2

+ p2c2,2.

The �rst-order conditions with respect to c2,1 and c2,2 give

p2(1− τ2) =U2(c2,1, c2,2)

U1(c2,1, c2,2)=

(c2,2

c2,1

)− 1σ

⇒ p2 =1

1− τ2

(c2,2

c2,1

)− 1σ

.

(4.14)Assuming that all tari� revenues are paid back lump sum to the

domestic households, the lump-sum transfers are

T1 =τ1

1− τ1

p2c1,2 and T2 =τ2

1− τ2

c2,1.


Substituting these in the budget constraints gives

a1 = c1,1 + p2c1,2

p2a2 = c2,1 + p2c2,2.

Since the trading costs do not result in any loss of resources, it followsthat the aggregate resource constraints are liai = l1c1,i + l2c2,i for bothi. Substituting for p2 from (4.13) and (4.14) in the budget constraints ofcountry 1 and 2, respectively, and combining with the aggregate resourceconstraints, gives the following characterization of the equilibrium:

a1 = c1,1 + (1− τ1)c1σ1,1c

σ−1σ

1,2 (4.15)

a2 = (1− τ2)c1σ2,2c

σ−1σ

2,1 + c2,2 (4.16)

a1 = c1,1 +l2l1c2,1 (4.17)

a2 =l1l2c1,2 + c2,2. (4.18)

From these, it immediately follows that

c2,1 =l1l2

(1− τ1)c1σ1,1c

σ−1σ

1,2 (4.19)

c1,2 =l2l1

(1− τ2)c1σ2,2c

σ−1σ

2,1 . (4.20)

Therefore, equations (4.15)-(4.18) implicitly determine ci,j as func-tions of a1, a2, τ1 and τ2.

Changing productivities and tari�s

I can now look at the e�ects of changes in a1, a2, τ1 and τ2. To see,for example, what is the e�ect of changing a1, I treat ci,j, given by theequilibrium conditions (4.15)-(4.18), as functions of a1, di�erentiate theequilibrium conditions implicitly with respect to a1 and solve for thederivatives of ci,j with respect to a1. The derivatives with respect to a2,τ1 and τ2 can be found in the same way. I perform these calculations inappendix 4.A.1.

Given all these derivatives, the changes in the consumption of theaggregate consumption good in country i, as x ∈ {a1, a2, τ1, τ2} changes,can be calculated as

∂ci∂x

=

(cici,1

) 1σ ∂ci,1∂x

+

(cici,2

) 1σ ∂ci,2∂x

. (4.21)


I calculate this for each of the four parameters in appendix 4.A.1.The results for changes in a1 are

∂c1

∂a1

=

(c1

c1,1

) 1σ

a2

c2,2+ σ−1

σ

(1 +

(a1

c1,1− 1)

11−τ1

)a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1)

∂c2

∂a1

=

(c2

c2,2

) 1σ l1l2

c1,2

c1,1

1σ

11−τ2

a2

c2,2+ τ2

1−τ2σ−1σ

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1) .

The sign of the �rst derivative is ambiguous. If σ ≥ 1, it is positive. Ifτ1 = 0, this can be shown to be equivalent to the condition in (4.12).The second derivative is unambiguously positive since a2

c2,2≥ 1 ≥ τ2.

For changes in a2, the changes in aggregate consumption are givenby

∂c1

∂a2

=

(c1

c1,1

) 1σ l2l1

c2,1

c2,2

1σ

11−τ1

a1

c1,1+ τ1

1−τ1σ−1σ

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1)

∂c2

∂a2

=

(c2

c2,2

) 1σ

a1

c1,1+ σ−1

σ

(1 +

(a2

c2,2− 1)

11−τ2

)a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1) .

Here, the �rst derivative is unambiguously positive. This means thatthe result from section 4.2.4, that a decrease in the productivity in coun-try 2 decreases welfare in country 1, holds also with trading costs. Thesign of the second derivative is ambiguous in the same way as was the�rst derivative with respect to c1.

For changes in τ1, the changes in aggregate consumption are givenby

∂c1

∂τ1

=

(c1

c1,1

) 1σ l2l1

1σa2

c2,2+ 1−σ

στ1

1−τ11σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c2,1

1− τ1

∂c2

∂τ1

=−(c2

c2,2

) 1σ l1l2

11−τ2

1σa2

c2,2+ τ2

1−τ2σ−1σ

1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c1,2

1− τ1

.

The sign of the �rst derivative is ambiguous. A su�cient conditionfor it to be positive is 1

σ> σ−1

στ1

1−τ1 ⇔ τ1 <1σ. The second derivative is

unambiguously negative since a2

c2,2≥ 1 ≥ τ2.

For changes in τ2, the changes in aggregate consumption are givenby


∂c1

∂τ2

=−(c1

c1,1

) 1σ l2l1

11−τ1

1σa1

c1,1+ τ1

1−τ1σ−1σ

1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c2,1

1− τ2

∂c2

∂τ2

=

(c2

c2,2

) 1σ l1l2

1σa1

c1,1+ 1−σ

στ2

1−τ21σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c1,2

1− τ2

Here, the �rst derivative is unambiguously negative while the sign ofthe second derivative is ambiguous.

So if one of the countries increases its tari�, this always decreaseswelfare in the other country while the e�ect on the country's own welfareis ambiguous.

Endogenizing climate

Consider now the situation where the changes in productivity are causedby the emissions of greenhouse gases made by country 1. An increase inemissions translates into a decrease in productivity in country 2. Emis-sions E are valued by country 1 as V (E). The total utility of country 1is then1

V (E) + u(c1),

where V is assumed to be such that

V ′ > 0 and V ′′ < 0. (4.22)

For given productivities and tari�s, c1 is determined competitivelyaccording to the conditions (4.15)-(4.18). Assume further that a1 isexogenously given and that a2 is a function of E.

I will make the following assumption about the productivity in coun-try 2 as a function of E:

da2

dE< 0 and

d2a2

dE2≤ 0. (4.23)

The equilibrium aggregate consumption in country 1, c1, is now afunction of τ1, τ2 and E. For given tari�s, the optimal choice of E forcountry 1 is given by

maxE

V (E) + u(c1).

1One motivation for this could be a two-period model where fossil fuels are onlyused in the �rst period and only by country 1. In the �rst period, productivity isgiven but in the second period, productivity depends on the �rst-period fossil-fueluse.


The �rst-order condition is

V ′(E) + u′(c1)∂c1

∂a2

da2

dE= 0. (4.24)

This condition implicitly de�nes the amount of emissions made bycountry 1 for given tari�s.

Consider now how the choice of E would change if one of the tari�s,τi, were to change. Imposing the optimality condition (4.24), E is nowa function of τ1 and τ2. Di�erentiating the optimality condition (4.24)implicitly with respect to τi gives

0 =V ′′(E)∂E

∂τi+ u′′(c1)

[∂c1

∂τi+∂c1

∂a2

da2

dE

∂E

∂τi

]∂c1

∂a2

da2

dE

+u′(c1)

[∂2c1

∂τi∂a2

+∂2c1

∂a22

da2

dE

∂E

∂τi

]da2

dE+ u′(c1)

∂c1

∂a2

d2a2

dE2

∂E

∂τi.

Solving for ∂E∂τi

gives

∂E

∂τi= −

u′′(c1)∂c1∂τi

∂c1∂a2

+ u′(c1) ∂2c1∂τi∂a2

V ′′(E) +

[u′′(c1)

(∂c1∂a2

)2

+ u′(c1)∂22c1∂a2

2

] (da2

dE

)2+ u′(c1) ∂c1

∂a2

d2a2

dE2

da2

dE.

To simplify the analysis, I will assume that the tari�s are small sothat I can evaluate all derivatives for τ1 = τ2 = 0. Given assumptions(4.23) and (4.24), and using that ∂c1

∂a2> 0, the �rst and third terms are

both negative. The second term (see appendix 4.A.2 for calculations) is

u′′(c1)

(∂c1

∂a2

)2

+ u′(c1)∂2c1

∂a22

= u′(c1)∂c1

∂a2

1

c2,2

1−θ−σσ

(a1

c1,1

)2

− 1σa1

c1,1

a2

c2,2(a1

c1,1

a2

c2,2

)2 .

A su�cient condition for this to be negative is that that θ ≥ 1. I willassume that the denominator is negative. Combined with da2

dE< 0, as

assumed in (4.23), this implies that

Sgn

(∂E

∂τi

)= Sgn

(−u′′(c1)

∂c1

∂τi

∂c1

∂a2

− u′(c1)∂2c1

∂τi∂a2

).

For τ1 (see appendix 4.A.2), we obtain

u′′(c1)∂c1

∂τ1

∂c1

∂a2

+u′(c1)∂2c1

∂τ1∂a22

= u′(c1)∂c1

∂τ1

1

c2,2

2−θ−σσ

(a1

c1,1

)2

+ σ−1σ

a1

c1,1

a2

c2,2(a1

c1,1

a2

c2,2

)2 .


When τ1 = 0, ∂c1∂τ1

> 0 and

Sgn

(∂E

∂τ1

)= Sgn

(θ + σ − 2

σ

a1

c1,1

+1− σσ

a2

c2,2

).

For τ2 (see appendix 4.A.2), we arrive at

u′′(c1)∂c1

∂τ2

∂c1

∂a2

+ u′(c1)∂2c1

∂τ2∂a22

= u′(c1)∂c1

∂τ2

1

c2,2

1−θσ

(a1

c1,1− 1)

+ 2σ−1σ

a1

c1,1

a2

c2,2

.

When τ1 = 0, ∂c1∂τ2

< 0 and

Sgn

(∂E

∂τ2

)= Sgn

(1− θσ

(a1

c1,1

− 1

)+ 2

σ − 1

σ

).

These results can be summarized in the following proposition:

Proposition 4.2. Assume that θ ≥ 1 and that the tari�s are small(τ1 ≈ τ2 ≈ 0). Then, if the emissions of greenhouse gases are given by(4.24), they depend on the tari�s as

Sgn

(∂E

∂τ1

)= Sgn

(θ + σ − 2

σ

a1

c1,1

+1− σσ

a2

c2,2

)Sgn

(∂E

∂τ2

)= Sgn

(1− θσ

(a1

c1,1

− 1

)+ 2

σ − 1

σ

)Proof. Follows from the above calculations.

The result for changes in τ2 implies that, if e.g. θ ≥ 1 and σ ≤ 1,country 2 could induce country 1 to reduce its emissions by threateningwith tari�s in the future. Since country 2 would gain from imposingtari�s, the threat would also be credible. Naturally, there are manyother strategic considerations that go into the setting of tari�s. Notethat, without further considerations, the credibility problem is ratherthat it would not be credible for country 2 to promise not to imposetari�s if country 1 reduces its emissions.

4.2.6 Climate change mitigation policy

The above suggested interpretation of the changes in productivity wasthat they were caused by climate change. An alternative interpretationis that productivity changes due to regulations to reduce greenhousegas emissions. That is, productivity decreases due to climate changemitigation policy. There should be similar e�ects from that kind ofchanges in productivity. So, the general equilibrium e�ects from trade

4.3. INSURANCE AGAINST WEATHER VARIABILITY 167

should also be included in calculations of the costs of mitigation policy.Proposition 4.1 implies that mitigation policy that reduces productivityin the north will decrease welfare in the south. As for the climate relatede�ects, the proposition does not rule out that a mitigation policy couldincrease welfare in the north. While there could still be adverse e�ectsfrom the non-traded sector, the other adverse e�ects of climate changewould not be present here.

This is the situation with which the carbon leakage literature is con-cerned and carbon leakage could provide counteracting bene�ts for un-regulated countries. Carbon leakage can occur in di�erent ways. Oneway is through the fossil-fuel price. If regulation in some countries de-creases the world market fossil-fuel price, this should increase the fossil-fuel use in other countries and it should bene�t those countries. Adi�erent way in which carbon leakage can materialize is if productionis relocated to countries with a less stringent regulation. To the extentthat this occurs, this should bene�t those countries to which produc-tion moves. In terms of the many countries model, countries to whichproduction can move could be interpreted as producing the same goodas the countries that production moves from and proposition 4.1 thengives that the indirect e�ects are positive for these countries. The em-pirical pollution haven literature (see Copeland and Taylor, 2004, foran overview) typically �nds that other factors are more important thanregulation for production location decisions. This seems to suggest thatthere should be limited bene�ts from production relocation in unregu-lated countries.

4.3 Insurance against weather variability

I will now, instead, consider the e�ects of changes in the distribution ofweather related shocks. Climate change is predicted to not just increaseaverage temperatures but also to change the distribution of weatheroutcomes. In particular, extreme weather events are predicted to becomemore frequent and more severe (IPCC 2007). Since these shocks will notbe perfectly correlated across the world, there will be scope for insuringagainst such shocks through trade in �nancial instruments. Since thedistribution of weather outcomes will depend on the amount of climatechange, so will the world market prices of the �nancial instruments thatcan be used to insure against these shocks. Therefore, as in the case oftrade in goods, countries will be indirectly a�ected by the changes in theweather distribution, and these indirect e�ects will go through changesin world market prices.

I will analyze these e�ects in the context of a many-country, two-period, endowment model. The �rst-period endowments are determin-


istic while the second-period endowments are stochastic. In the �rstperiod, there is trade in a complete set of �nancial instruments withstate contingent second-period payo�s.

In section 4.3.1 I will set up the model. In section 4.3.2 I will solvefor the equilibrium allocation. I will then, in section 4.3.3, consider howchanges in the endowment distribution a�ect welfare. In section 4.3.4, Iwill consider what the general results imply for the two-country (northand south) case.

4.3.1 Model setup

Consider a two-period model. There are I countries and country i haspopulation li. In the �rst period, the representative household in eachcountry i receives a deterministic endowment y1,i. In the second pe-riod, there are N possible states. In state n, the representative house-hold receives an endowment y2,i,n. The probability of the second-periodstate being n is πn, where

∑Nn=1 πn = 1. Changes in the distribution of

weather-induced shocks will be modeled as changes in {πn} and {y2,i,n}.Let

Y1 =I∑i=1

liy1,i and Y2,n =I∑i=1

liy2,i,n

andsi,n =

y2,i,n

Y2,n

. (4.25)

Y1 is the total endowment in period 1. Y2,n is the total endowment inthe second-period state n and si,n is the share of that endowment thatgoes to the representative household in country i, that is, the share oftotal endowment in state n that the representative household in countryi receives.

Let the consumption of the representative household in country i bec1,i in period 1 and c2,i,n in period 2 if the state is n. Households valuean allocation as the discounted expected utility from consumption:

Vi = u(c1,i) + βE1 [u(c2,i,n)] = u(c1,i) + βN∑n=1

πnu(c2,i,n), (4.26)

where

u(c) =c1−θ − 1

1− θ. (4.27)

There are complete markets for insurance against the second-perioduncertainty. For each state n, there is an asset that pays one unit ofconsumption in the second period if the state is n and 0 in all otherstates. Let bi,n be the holdings of the representative household in country


i of the asset that pays in state n and let qn be the price, in period 1, ofthe asset that pays in state n.

The asset holdings must ful�ll the market clearing condition:

I∑i=1

libi,n = 0 for all n. (4.28)

Consumption is given by

c1,i = y1,i −N∑n=1

qnbi,n and c2,i,n = y2,i,n + bi,n. (4.29)

4.3.2 Equilibrium

I will now solve for the equilibrium allocation. Households are assumedto act as price takers and buy assets to maximize discounted expectedutility. As in the case of trade in goods, the price-taking assumption isimportant since the indirect e�ects go through changes in prices.

An equilibrium is a set of prices {qn}Nn=1 and asset holdings {bi,n}i,nsuch that

• households maximize utility (4.26) with consumption given by(4.29)

• the asset holdings ful�ll the market clearing condition (4.28) forall n

The optimization problem of the representative household in countryi is

max{bi,n}Nn=1

u

(y1,i −

N∑n=1

qnbi,n

)+ β

N∑n=1

πnu (y2,i,n + bi,n) .

In principle, there should be non-negativity constraints on consump-tion. However, since limc→0 u

′(c) = ∞, the non-negativity constraintswill never bind.

The �rst-order condition with respect to bi,n is

qnu′(c1,i) = βπnu

′(c2,i,n)⇒ qn = βπnu′(c2,i,n)

u′(c1,i). (4.30)

In this expression, only the ratio of marginal utilities depends on i.Thus, this ratio will be the same for all countries. The representativehousehold balances the price of the assets against the marginal value ofconsumption in period 1 and in di�erent states of period 2, taking thetime preference and the realization probabilities into account.


Using the functional form of the utility function (4.27), we obtain

qn = βπnu′(c2,i,n)

u′(c1,i)= βπn

(c2,i,n

c1,i

)−θ⇒ c2,i,n =

(βπnqn

) 1θ

c1,i.

Substituting the expressions for consumption from (4.29), multiplyingby li and summing over i on both sides gives

I∑i=1

li (y2,i,n + bi,n) =I∑i=1

(βπnqn

) 1θ

li

(y1,i −

N∑n′=1

qn′bi,n′

)I∑i=1

liy2,i,n +I∑i=1

libi,n =

(βπnqn

) 1θ

I∑i=1

liy1,i −(βπnqn

) 1θ

N∑n′=1

qn′I∑i=1

libi,n′

I∑i=1

liy2,i,n =

(βπnqn

) 1θ

I∑i=1

liy1,i,

where the market clearing condition for asset holdings (4.28) was usedfor the last step.

Solving for qn gives

qn = βπn

( ∑Ii=1 liy1,i∑Ii=1 liy2,i,n

)θ

= βπn

(Y1

Y2,n

)θ. (4.31)

We see that the price will be higher for assets that pay in states that arerealized with high probability and states where the total endowment issmall.

Comparing this to the �rst-order condition with respect to bi,n in(4.30) gives that, for all i and n,

c1,i

c2,i,n

=Y1

Y2,n

⇒ c2,i,n =c1,i

Y1

Y2,n.

Now de�nesi =

c1,i

Y1

;

this is the share of the total endowment that the representative house-hold in country i consumes in the �rst period and in each state in thesecond period.

From the consumption expressions (4.29), we see that

siY1 = c1,i = y1,i −N∑n=1

qnbi,n ⇒ y1,i − siY1 =N∑n=1

qnbi,n


and that

siY2,n = c2,i,n = y2,i,n + bi,n ⇒ bi,n = siY2,n − y2,i,n.

Combining these expressions gives

y1,i − siY1 =∑N

n=1 qn (siY2,n − y2,i,n)

⇒ yi,1 +∑N

n=1 qny2,i,n = si

(Y1 +

∑Nn=1 qnY2,n

).

Thus, we arrive at

si =yi,1 +

∑Nn=1 qny2,i,n

Y1 +∑N

n=1 qnY2,n

.

We see that si is also the wealth share of the representative householdin country i. The share can be rewritten, in terms of exogenous objects,as

si =yi,1 +

∑Nn=1 βπn

Y θ1Y θ2,n

y2,i,n

Y1 +∑N

n=1 βπnY θ1Y θ2,n

Y2,n

=yi,1 + Y θ

1

∑Nn=1 βπnY

−θ2,n y2,i,n

Y1 + Y θ1

∑Nn=1 βπnY

−θ2,n Y2,n

=

yi,1Y1Y 1−θ

1 + β∑N

n=1 πny2,i,n

Y2,nY 1−θ

2,n

Y 1−θ1 + β

∑Nn=1 πnY

1−θ2,n

.

Using the second-period endowment share de�ned in (4.25), the wealthshare becomes

si =

yi,1Y1Y 1−θ

1 + β∑N

n=1 πnsi,nY1−θ

2,n

Y 1−θ1 + β

∑Nn=1 πnY

1−θ2,n

. (4.32)

For given Y1 and Y2,n, the wealth share is always increasing in y1,i

and y2,i,n. That is, a higher endowment for the representative householdis always preferable given total endowments.

Assume that Y 1−θ1 +β

∑Nn=1 πnY

1−θ2,n is given. The wealth share, si, is

increasing in each endowment share, si,n, but, comparing states, whichstates the country would like to have high shares in depends on the valueof θ. If θ > 1, the country prefers to have a large share of the endowment,si,n, in states n where Y2,n is small. If θ < 1 the opposite holds. Having alarge share in a state with a high endowment gives a higher endowment,but the value of that endowment is smaller since consumption in thatstate is valued less. Which of these e�ects that dominates depends on θ.

Using the shares, consumption can be written

c1,i = siY1 and c2,i,n = siY2,n.


The equilibrium asset holdings are

c2,i,n = y2,i,n + bi,n ⇒ siY2,n = si,nY2,n + bi,n.

Solving for the asset holdings gives

bi,n = (si − si,n)Y2,n. (4.33)

Thus, the representative household holds positive amounts of the assetsthat pay in states where the endowment share is smaller than the wealthshare, and the other way around. Note that it is possible for a countryto hold negative (or positive) amounts of all assets if much (little) of thewealth comes from the period 1 endowment y1,i.

In conclusion, the equilibrium is determined by the asset prices (4.31)and the asset holdings (4.33) with endowment shares and wealth sharesgiven by (4.25) and (4.32), respectively. Using these, the allocation ofconsumption can be calculated from (4.29).

4.3.3 Welfare e�ects of changing the second-period

distribution

Starting from the equilibrium allocation for a given distribution, that isgiven values of πn, y1,i and y2,i,n, we can now look at how changes inthe distribution a�ect welfare. A change in the probabilities, πn, can beconsidered as a change in the probability of an event such as a heatwaveor a storm occurring. A change in the endowment in a given state, y2,i,n,can be considered as a change in the severity of such an event. To beable to distinguish more easily between the direct e�ects and the e�ectsthrough changing equilibrium prices, we can start from an expressionwith consumption from (4.29) substituted in the expression for welfare(4.26), namely,

Vi = u

(y1,i −

N∑n=1

qnbi,n

)+ β

N∑n=1

πnu(y2,i,n + bi,n),

where prices qn and asset holdings bi,n are at their equilibrium values.Consider now a change ∆ that a�ects probabilities and second-periodendowments. Since asset prices and asset holdings depend on the en-dowment distribution, these will change endogenously in response to the


change ∆. The change in welfare is given by

dVid∆

=−u′(c1,i)N∑i=1

(dqnd∆

bi,n + qndbi,nd∆

)+ β

N∑n=1

dπnd∆

u(c2,i,n)

+βN∑n=1

πnu′(c2,i,n)

(dy2,i,n

d∆+dbi,nd∆

)

=−u′(c1,i)N∑i=1

dqnd∆

bi,n + β

N∑n=1

dπnd∆

u(c2,i,n) + β

N∑n=1

πnu′(c2,i,n)

dy2,i,n

d∆

+N∑n=1

(βπnu′(c2,i,n)− u′(c1,i)qn)

dbi,nd∆

.

Using the �rst-order condition with respect to bi,n (4.30), the changein welfare is

dVid∆

= βN∑n=1

dπnd∆

u(c2,i,n) + βN∑n=1

πnu′(c2,i,n)

dy2,i,n

d∆− u′(c1,i)

N∑n=1

dqnd∆

bi,n.

(4.34)The e�ect of the change ∆ on welfare can be divided into two direct

e�ects, the �rst two e�ects in the last expression, and an indirect pricee�ect.

The direct e�ect of the changes in probabilities is such that a redis-tribution of the probability mass from states with low consumption tostates with high consumption increases welfare. Note that the changesin probabilities must sum to 0. For all countries, the states with highconsumption are the states with a high combined endowment, Y2,n.

The direct e�ect of changes in endowments is such that an increasein the endowment in any state increases welfare.

The price e�ect is such that welfare increases from price increases(decreases) of assets of which the household holds negative (positive)amounts. From (4.33), the household holds positive (negative) amountsof the assets that pay in states where the household's endowment shareis smaller (larger) than the wealth share. From the expression for theprices (4.31), it follows that prices of the asset that pays in state nincrease in πn and decrease in Y2,n.

To see what is the total e�ect, I will consider changes in probabilitiesand endowments separately.

Changes in realization probabilities

Start by assuming that dy2,i,n

d∆= 0 for all i and n. Using the expression

for prices (4.31), the welfare e�ect given by (4.34) becomes


dVid∆

= β

N∑n=1

dπnd∆

u(c2,i,n)− u′(c1,i)N∑n=1

dqnd∆

bi,n

= β

N∑n=1

dπnd∆

u(c2,i,n)− u′(c1,i)N∑n=1

qnπn

dπnd∆

bi,n

=N∑n=1

[βu(c2,i,n)− u′(c1,i)

qnπnbi,n

]dπnd∆

=N∑n=1

[βu(c2,i,n)− βu′(c2,i,n)bi,n]dπnd∆

= βN∑n=1

[u (siY2,n)− u′ (siY2,n) (si − si,n)Y2,n]dπnd∆

.

For logarithmic utility, θ = 1, this expression becomes

dVid∆

= βN∑n=1

[log (siY2,n)− (si − si,n)Y2,n

siY2,n

]dπnd∆

= βN∑n=1

[log (Y2,n)− (si − si,n)

si

]dπnd∆

+ β log(si)N∑n=1

dπnd∆

= βN∑n=1

[log (Y2,n) +

si,nsi− 1

]dπnd∆

= β

N∑n=1

[log (Y2,n) +

si,nsi

]dπnd∆

. (4.35)

We see that the representative household would prefer a redistributionof the probability mass to states with a higher total endowment Y2,n anda higher endowment share si,n.


For θ 6= 1, the expression becomes

dVid∆

= β

N∑n=1

[c1−θ

2,i,n − 1

1− θ− c−θ2,i,nbi,n

]dπnd∆

= β

N∑n=1

[(siY2,n)1−θ

1− θ− (siY2,n)−θ (si − si,n)Y2,n

]dπnd∆

(4.36)

− β

1− θ

N∑n=1

dπnd∆

= β

N∑n=1

(siY2,n)1−θ[

1

1− θ− si − si,n

si

]dπnd∆

= βs1−θi

N∑n=1

Y 1−θ2,n

[1

1− θ− si − si,n

si

]dπnd∆

= βs1−θi

N∑n=1

Y 1−θ2,n

[θ

1− θ+si,nsi

]dπnd∆

. (4.37)

Here, the welfare e�ects are more complicated. Comparing two stateswith the same total endowment Y2,n, a state with a higher endowmentshare si,n is better. If θ < 1, states with a larger total endowment Y2,n

are better. If θ > 1, a larger Y2,n is better (worse) if si,n is small (large)enough so that the parenthesis is negative (positive). If the endowmentshare is large, the household can bene�t from a small total endowmentsince that would increase the price of assets that the household sells.

This can all be summarized in the following proposition:

Proposition 4.3. For logarithmic utility, θ = 1, the welfare of the rep-resentative household in country i is increased if the probability mass is

redistributed towards second-period states n such that[log (Y2,n) +

si,nsi

]is large. For θ 6= 1, the welfare of the representative household in countryi is increased if the probability mass is redistributed towards second-period

states n such that Y 1−θ2,n

[θ

1−θ +si,nsi

]is large.

Proof. Follows from (4.35) and (4.37).

This concludes the description of the welfare e�ects of varying therealization probabilities {πn}Nn=1.

Changes in second-period endowments

Consider now instead the case where dπnd∆

= 0 for all n. The e�ects ofchanges in endowments can be analyzed for an individual endowment.So I will consider a change in the endowment y2,i′,n.


From (4.31), the price change induced by a change in y2,i′,n is

dqndy2,i′,n

= −θ qnY2,n

dY2,n

dy2,i′,n= −θ qn

Y2,n

li′ .

Using this in (4.34), the welfare e�ect, for the representative house-hold in country i, from a change in y2,i′,n is

dVidy2,i′,n

= βπnu′(c2,i,n)

dy2,i,n

dy2,i′,n− u′(c1,i)

dqndy2,i′,n

bi,n

= βπnu′(c2,i,n)

dy2,i,n

dy2,i′,n+ u′(c1,i)θ

qnY2,n

li′bi,n.

The �rst, direct, e�ect is positive if i′ = i and zero otherwise. Thesecond, indirect, e�ect has the same sign as bi,n. If y2,i′,n increases, thisdecreases the price qn and this is positive or negative depending on thesign of bi,n. So, if i′ 6= i, the total e�ect only depends on bi,n. If i′ = i,we see that

dVidy2,i,n

= βπnu′(c2,i,n) + θu′(c1,i)

qnY2,n

libi,n = {FOC} =

= βπnu′(c2,i,n) + θβπnu

′(c2,i,n)bi,nY2,n

li

= βπnu′(c2,i,n)

(1 + θ

(si − si,n)Y2,n

Y2,n

li

)= βπnu

′(c2,i,n) (1 + θ(si − si,n)li) .

The derivative can be negative if θ is large and si,n is larger than si.However, quantitatively, this does not seem very likely since it wouldrequire a large di�erence between si and si,n. If, e.g., θ = 2 it wouldrequire (si − si,n)li > 0.5.

This discussion is summarized in the following proposition:

Proposition 4.4. The welfare e�ects for the representative householdin country i from a change in y2,i′,n satisfy

Sgn

(dVi

dy2,i′,n

)=

{Sgn(bi,n) if i′ 6= iSgn (1 + θ(si − si,n)li) if i

′ = i

with bi,n given by (4.33), si given by (4.32) and si,n given by (4.25).

Proof. Follows from the calculations above.

This concludes the description of the welfare e�ects of changes in{y2,i,n}.

4.4. CONCLUSIONS AND DISCUSSION 177

4.3.4 The two-country case

Consider now the two-country case similar to that discussed in section4.2.4. It is not possible to have changes in the realization probabilitiesthat only a�ect country 2. So I will only consider changes in the second-period endowments in country 2. The welfare e�ects in country 1 are

dV1

d∆=−u′(c1,1)

N∑n=1

dqnd∆

b1,n = u′(c1,1)N∑n=1

θqnY2,n

l2b1,ndy2,2,n

d∆

=u′(c1,1)θl2

N∑n=1

qnY2,n

b1,ndy2,2,n

d∆

The sign of the e�ect will depend on the pattern of changes in endow-ments. One prediction is that climate change will increase the severity ofextreme events. If this is interpreted as that endowments will decreasethe most in states where the endowments y2,2,n are small to begin with,it seems likely that these will be states where b1,n < 0 since country 2would want to buy an insurance against these outcomes. This wouldmake the welfare e�ect on country 1 positive.

So, the insurance instruments channel would tend to make country1 less interested in reducing the emissions of greenhouse gases.

4.4 Conclusions and discussion

This chapter has highlighted two ways in which countries are economi-cally linked to each other and how these links a�ect the calculations ofgains and losses associated with climate change. These channels implythat a country that is not directly a�ected by climate change will stillindirectly be a�ected through changes in world market prices. The gen-eral conclusion is that if a country is a net seller of a good (or a �nancialinstrument), changes that increase the demand or decrease the supplyof this good will bene�t the country since this will increase the relativevalue of the goods that the country sells. For goods of which the countryis a net buyer, the opposite will hold. In the stylized two-country exam-ple, this implies that looking at the channel through trade in goods, thenorth would be hurt by decreased productivity in the south. When look-ing at trade in �nancial instruments that can be used to insure againstweather variability, the north will tend to gain if the severity of theextreme bad weather events in the south increases.

In both these cases, all agents have been assumed to be price takers.This assumption is important since the indirect e�ects go through chang-ing world market prices. When looking at trade in goods, a country that


is negatively a�ected by climate change will also experience an o�settinge�ect since the relative price of the goods that it produces will increase.This o�setting e�ect captures the potential to increase income throughmark-up pricing. So, if prices are not competitively set to begin with,these indirect e�ects will be di�erent. This is something that would beinteresting to investigate further.

In the parts on insurance, I have assumed complete markets. Thisis not a particularly realistic assumption. Arrow et al. (1996) pointout that insurance companies may be reluctant to sell insurance againstevents with unknown probability distributions. While this may very wellbe true, the instruments discussed here need not literally be insurance.A di�erent version of the model with endogenous investments in capitalcould be set up. In that model, the second-period distribution could beon productivity of capital in the countries. Such a model gives similarresults and allowing for direct investments in capital in other countriesgives complete markets. Chichilnisky (1998) argues that if there is un-certainty about the future distribution of events, this will mean that in-struments contingent on these events will not provide complete markets.There are also e�ects of climate change that are not tied to productiv-ities of capital. So, it might be reasonable to assume that markets arenot complete. My guess is, however, that this would only be problematicfor the results derived here if there is a systematic relationship betweenmarket imperfections and the e�ects of climate change.

For tractability and to highlight the basic mechanisms, some impor-tant aspects have been omitted from the analysis. In the model withtrade in goods, the comparative advantages were assumed to be exoge-nously given and regardless of climate change, the comparative advan-tages are strong enough to induce the countries to specialize in one givengood. To the extent that the comparative advantages are still such thatthe signs of net exports of the goods are the same, regardless of climatechange, this does not seem very problematic. The indirect e�ects goingthrough prices will still be such that price increases (decreases) bene-�t countries that are net exporters (importers) of the good. One casewhere this might not be the case is agricultural productivity. The grow-ing zones for various crops will tend to move away from the equator. Sofor some goods, the comparative advantages may change directions. Thissuggests that having good-speci�c productivities, endogenous choices ofwhat to produce and considering changes in the good-speci�c produc-tivities could add an interesting structure to the indirect e�ects.

Another aspect that has been completely omitted from the analy-sis is that the e�ects of climate change are endogenous. In the sectionon trade with trade barriers, the amount of climate change was endo-

4.4. CONCLUSIONS AND DISCUSSION 179

genized. However, for a given amount of climate change, the e�ectsof this change can also be endogenous. By adapting to future climatechange, a country can become less vulnerable to future variability. Sim-ilarly, a country could try to adapt its production possibilities towardsthe production of goods where the country can be expected to have acomparative advantage in the future. How this would a�ect the calcula-tions is di�cult to say without modeling it. One implication would bethat expectations of future climate change would decrease the resourcesavailable today, since some of these resources would be used for adaptingto future climate change. In terms of the two-country model with tradein goods, this could be modeled with two periods where the productivityin the south in the �rst period would decrease if there were expectationsof climate change. In such a model, the same negative indirect e�ectsfor the north would then be present in the �rst period. In terms of themodel with trade in �nancial instruments, expectations of future climatechange would decrease the available endowments in the �rst period inthe countries that want to adapt to future changes. When looking attrade in goods and interpreting changes in productivity as the result ofmitigation policy, the exogeneity assumptions may be more reasonablesince the time span is shorter.

In this chapter, I used a Ricardian type of model for trade. This typeof model is not very well suited to capture the signi�cant volume of tradein similar goods that takes place between similar countries. Furthermore,a share of GDP is not traded at all. So an improvement of the modelcould be to include a non-traded sector. The countries could then beinterpreted as larger regions and the trade in similar goods among similarcountries would then belong to the non-traded sector. As argued above,this would probably not change the sign of the indirect e�ect on countriesthat are not directly a�ected by climate change, but it could have a largeimpact on the net e�ects in countries that are a�ected.

This chapter has illustrated two channels through which economiesare interconnected and shown that there will be indirect e�ects in addi-tion to the direct e�ects experienced by countries. There will likely alsobe other types of indirect e�ects of climate change. Commonly discussedexamples include migration and increased risks of con�icts. I leave theseand other aspects to future research.

In conclusion, I would like to reemphasize the general theme here:since countries are interconnected in many ways, any calculation of thee�ects of climate change that a country will experience should be basedon the total, general equilibrium e�ects.

References

Antweiler, W., B. R. Copeland & M. S. Taylor, 2001, �Is free trade goodfor the environment?�, American Economic Review 91(4): 877-908.

Arrow, K.J, J. Parikh & G. Pillet (Principal Lead Auhtors), 1996, �Decision-Making Frameworks for Addressing Climate Change�, in J. P. Bruce, H.Lee and E. F. Haites (Eds) Climate Change 1995 - Economic and SocialDimensions of Climate Change: Contribution of Working Group III tothe Second Assessment Report of the Intergovernmental Panel on Cli-mate Change, Cambridge University Press.

Chichilnisky, G., 1998, �The Economics of Global Environmental Risks.�,International Yearbook of Environmental and Resource Economics, Vol.II. eds. T. Tietenberg & H. Folmer, Edward Elgar: 235-273.

Copeland, B. R. & M. S. Taylor, 2004, �Trade growth and the environ-ment�, Journal of Economic Literature, 42(1): 7-71.

Di Maria, C. & S. A. Smulders, 2004, �Trade pessimists vs technologyoptimists: Induced technical change and pollution havens�, Advances inEconomic Analysis & Policy, 3 (2): Article7.

Di Maria, C. & E. van der Werf, 2008, �Carbon leakage revisited: Uni-lateral climate policy with directed technical change�, Environmental &Resource Economics, 39 (2): 55-74.

Golombek, R. & M. Hoel, 2004, �Unilateral emission reductions andcross-country technology spillovers�, Advances in Economic Analysis &Policy, 4 (2): Article 3.

Hemous, D., 2012, �Environmental Policy and Directed Technical Changein a Global Economy: Is There a Case for Carbon Tari�s?�, mimeo Har-vard University.

IPCC, 2007, Climate Change 2007: The Fourth Assessment Report ofthe Intergovernmental Panel on Climate Change[WG1], Cambridge Uni-versity Press, Cambridge.

Nordhaus, W. & J. Boyer, 2000, Warming the World: Economic Modelsof Global Warming, MIT Press, Cambridge, MA.

180

4.A. CALCULATIONS FOR TRADE IN GOODS 181

4.A Calculations for trade in goods with trading costs

In this appendix, I show the calculations for the two-country model fortrade in goods with import tari�s.

4.A.1 Comparative statics for changes in a1, a2, τ1

and τ2}

Consider a change in x ∈ {a1, a2, τ1, τ2}. This change will result inendogenous changes in equilibrium values of ci,j. Let primes denotederivatives with respect to x (for each choice of x, the correspondinga′1, a

′2, τ

′1 or τ ′2 will be one and the other derivatives will be zero) .

Di�erentiating the equilibrium conditions (4.15)-(4.18) with respect tox gives

a′1 =

(1 +

1

σ

l2l1

c2,1

c1,1

)c′1,1 +

σ − 1

σ

l2l1

c2,1

c1,2

c′1,2 −l2l1

c2,1

1− τ1

τ ′1

a′2 =σ − 1

σ

l1l2

c1,2

c2,1

c′2,1 +

(1 +

1

σ

l1l2

c1,2

c2,2

)c′2,2 −

l1l2

c1,2

1− τ2

τ ′2

a′1 = c′1,1 +l2l1c′2,1

a′2 =l1l2c′1,2 + c′2,2.

Setting x = a1, a′1 = 1 and a′2 = τ ′1 = τ ′2 = 0 gives

∂c1,1

∂a1

=

a2

c2,2+ σ−1

σ

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1)

∂c1,2

∂a1

=

σ−1σ

c1,2c1,1

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1)

∂c2,1

∂a1

=c2,1

c1,1

1 + 1σl1l2

c1,2c2,2

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1)

∂c2,2

∂a1

=l1l2

1−σσ

c1,2c1,1

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1) .


Setting a′2 = 1 and a′1 = τ ′1 = τ ′2 = 0 gives

∂c1,1

∂a2

=l2l1

1−σσ

c2,1c2,2

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1)

∂c1,2

∂a2

=c1,2

c2,2

1 + 1σl2l1

c2,1c1,1

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1)

∂c2,1

∂a2

=

σ−1σ

c2,1c2,2

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1)

∂c2,2

∂a2

=

a1

c1,1+ σ−1

σ

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1) .

Setting x = τ1, τ ′1 = 1 and a′1 = a′2 = τ ′2 = 0 gives

∂c1,1

∂τ1

=l2l1

1 + 1σl1l2

c1,2c2,2

1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c2,1

1− τ1

∂c1,2

∂τ1

=1−σσ

1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c1,2

1− τ1

∂c2,1

∂τ1

=−1 + 1

σl1l2

c1,2c2,2

1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c2,1

1− τ1

∂c2,2

∂τ1

=l1l2

σ−1σ

1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c1,2

1− τ1

.

Setting x = τ2, τ ′2 = 1 and a′1 = a′2 = τ ′1 = 0 gives

∂c1,1

∂τ2

=l2l1

σ−1σ

1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c2,1

1− τ2

∂c1,2

∂τ2

=−1 + 1

σl2l1

c2,1c1,1

1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c1,2

1− τ2

∂c2,1

∂τ2

=1−σσ

1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c2,1

1− τ2

∂c2,2

∂τ2

=l1l2

1 + 1σl2l1

c2,1c1,1

1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c1,2

1− τ2

.


Di�erentiating utility given by (4.2) with respect to x gives

∂ci∂x

u(ci) = u′(ci)∂

∂x,

where

∂ci∂x

=

(cici,1

) 1σ ∂ci,1∂x

+

(cici,2

) 1σ ∂ci,2∂x

.

When calculating this derivative for di�erent i and x, I use expressions(4.19) and (4.20). I also use that l2c2,1

l1c1,1= a1

c1,1− 1 and l1c1,2

l2c2,2= a2

c2,2− 1.

For changes in a1, the changes in aggregate consumption are givenby

∂c1

∂a1

=

(c1

c1,1

) 1σ

a2

c2,2+ σ−1

σ

(1 +

(a1

c1,1− 1)

11−τ1

)a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1)

∂c2

∂a1

=

(c2

c2,2

) 1σ l1l2

c1,2

c1,1

1σ

11−τ2

a2

c2,2+ τ2

1−τ2σ−1σ

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1) .

Note that the second derivative is unambiguously positive since a2

c2,2≥

1 ≥ τ2.For changes in a2, the changes in aggregate consumption are given

by

∂c1

∂a2

=

(c1

c1,1

) 1σ l2l1

c2,1

c2,2

1σ

11−τ1

a1

c1,1+ τ1

1−τ1σ−1σ

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1)

∂c2

∂a2

=

(c2

c2,2

) 1σ

a1

c1,1+ σ−1

σ

(1 +

(a2

c2,2− 1)

11−τ2

)a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1) .

For changes in τ1, the changes in aggregate consumption are given by

∂c1

∂τ1

=

(c1

c1,1

) 1σ l2l1

1σa2

c2,2+ 1−σ

στ1

1−τ11σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c2,1

1− τ1

∂c2

∂τ1

=−(c2

c2,2

) 1σ l1l2

11−τ2

1σa2

c2,2+ τ2

1−τ2σ−1σ

1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c1,2

1− τ1

.


For changes in τ2, the changes in aggregate consumption are given by

∂c1

∂τ2

=−(c1

c1,1

) 1σ l2l1

11−τ1

1σa1

c1,1+ τ1

1−τ1σ−1σ

1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c2,1

1− τ2

∂c2

∂τ2

=

(c2

c2,2

) 1σ l1l2

1σa1

c1,1+ 1−σ

στ2

1−τ21σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) c1,2

1− τ2

.

4.A.2 Calculations for endogenizing climate

The second-order partial derivatives of c1 are2

∂2c1

∂a22

=∂c1

∂a2

[1

σ

1

c1

∂c1

∂a2

− 1

σ

1

c1,1

∂c1,1

∂a2

+1

c2,1

∂c2,1

∂a2

− 1

c2,2

∂c2,2

∂a2

]

+∂c1

∂a2

− 1σ

11−τ1

a1

c21,1

∂c1,1∂a2

1σ

11−τ1

a1

c1,1+ τ1

1−τ1σ−1σ

−− a1

c21,1

∂c1,1∂a2

+ 1c2,2− a2

c22,2

∂c2,2∂a2

a1

c1,1+ a2

c2,2+ 1

σ

(c2,1c1,2c1,1c2,2

− 1)

∂2c1

∂a2∂τ1

=∂c1

∂τ1

1

σ

1

c1

∂c1

∂a2

− 1

σ

1

c1,1

∂c1,1

∂a2

+

1σ

(1c2,2− a2

c22,2

∂c2,2∂a2

)1σa2

c2,2+ 1−σ

στ1

1−τ1

+∂c1

∂τ2

− 1σ

(− a1

c21,1

∂c1,1∂a2

+ 1c2,2− a2

c22,2

∂c2,2∂a2

)1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) +

1

c2,1

∂c2,1

∂a2

∂2c1

∂a2∂τ2

=∂c1

∂τ2

1

σ

1

c1

∂c1

∂a2

− 1

σ

1

c1,1

∂c1,1

∂a2

+− 1σ

11−τ1

a1

c21,1

∂c1,1∂a2

1σ

11−τ1

a1

c1,1+ τ1

1−τ1σ−1σ

+∂c1

∂τ2

− 1σ

(− a1

c21,1

∂c1,1∂a2

+ 1c2,2− a2

c22,2

∂c2,2∂a2

)1σ

(a1

c1,1+ a2

c2,2

)+ 1

σ2

(c2,1c1,2c1,1c2,2

− 1) +

1

c2,1

∂c2,1

∂a2

.I will analyze the case where the tari�s are small and set τ1 = τ2 = 0.

Then, combining (4.13) and (4.14), c2,1c1,2c1,1c2,2

= 1 and we arrive at

c1,1

a1

= 1− c2,2

a2a1

c1,1

+a2

c2,2

=a1

c1,1

a2

c2,2

.

2Note thatc2,1c1,2c1,1c2,2

only depends on the tari�s and is therefore independent of a2.


The derivatives wrt a2 are

∂c1,1

∂a2

=l2l1

1−σσ

c2,1c2,2

a1

c1,1+ a2

c2,2

=c1,1

c2,2

σ−1σ

(a1

c1,1− 1)

a1

c1,1+ a2

c2,2

∂c2,1

∂a2

=c2,1

c2,2

σ−1σ

a1

c1,1+ a2

c2,2

∂c2,2

∂a2

=

a1

c1,1+ σ−1

σ

a1

c1,1+ a2

c2,2

∂c1

∂a2

=c1

c2,2

1

σ

a1

c1,1− 1

a1

c1,1+ a2

c2,2

.

Some useful combinations of derivatives are

1

c1

∂c1

∂a2

− 1

c1,1

∂c1,1

∂a2

=1

c2,2

a1

c1,1− 1

a1

c1,1+ a2

c2,2

1

c2,1

∂c2,1

∂a2

− 1

c2,2

a1

c1,1

∂c2,2∂a2

+ 1a1

c1,1+ a2

c2,2

=− 1

c2,2

1σa2

c2,2+(a1

c1,1

)2

+ a1

c1,1(a1

c1,1+ a2

c2,2

)2

1

c2,2

∂c2,2

∂a2

− 1

c1,1

∂c1,1

∂a2

=1

c2,2

a1

c1,1a1

c1,1+ a2

c2,2

2σ − 1

σ.

Using these, we obtain

∂2c1

∂a22

=∂c1

∂a2

1

c2,2

1−σσ

(a1

c1,1

)2

− 1σa1

c1,1

a2

c2,2(a1

c1,1+ a2

c2,2

)2

∂2c1

∂a2∂τ1

=∂c1

∂τ1

1

c2,2

2−σσ

(a1

c1,1

)2

+ σ−1σ

a1

c1,1

a2

c2,2(a1

c1,1+ a2

c2,2

)2

∂2c1

∂a2∂τ2

=∂c1

∂τ2

1

c2,2

1σ

(a1

c1,1− 1)

+ 2σ−1σ

a1

c1,1+ a2

c2,2


and

u′′(c1)

(∂c1

∂a2

)2

+ u′(c1)∂2c1

∂a22

=u′(c1)∂c1

∂a2

1

c2,2

1−θ−σσ

(a1

c1,1

)2

− 1σa1

c1,1

a2

c2,2(a1

c1,1

a2

c2,2

)2

u′′(c1)∂c1

∂τ1

∂c1

∂a2

+ u′(c1)∂2c1

∂τ1∂a22

=u′(c1)∂c1

∂τ1

1

c2,2

2−θ−σσ

(a1

c1,1

)2

+ σ−1σ

a1

c1,1

a2

c2,2(a1

c1,1

a2

c2,2

)2

u′′(c1)∂c1

∂τ2

∂c1

∂a2

+ u′(c1)∂2c1

∂τ2∂a22

=u′(c1)∂c1

∂τ2

1

c2,2

1−θσ

(a1

c1,1− 1)

+ 2σ−1σ

a1

c1,1

a2

c2,2

.

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47. Kohlscheen, Emanuel Essays on Debts and Constitutions, 2004

48. Olovsson, Conny Essays on Dynamic Macroeconomics, 2004

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190 MONOGRAPH SERIES

50. Herzing, Mathias Essays on Uncertainty and Escape in Trade Agree-ments, 2005

51. Bon�glioli, Alessandra Essays on Financial Markets and Macroe-conomics, 2005

52. Pienaar, Natalie Economic Applications of Product Quality Regu-lation in WTO Trade Agreements, 2005

53. Song, Zheng Essays on Dynamic Political Economy, 2005

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55. Favara, Giovanni Credit and Finance in the Macroeconomy, 2006

56. Björkman, Martina Essays on Empirical Development Economics:Education, Health and Gender, 2006

57. Larsson, Anna Real E�ects of Monetary Regimes, 2007

58. Prado, Jr., Jose Mauricio Essays on Public Macroeconomic Policy,2007

59. Tonin, Mirco Essays on Labor Market Structures and Policies, 2007

60. Queijo von Heideken, Virgina Essays on Monetary Policy and As-set Markets, 2007

61. Finocchiaro, Daria Essays on Macroeconomics, 2007

62. Waisman, Gisela Essays on Discrimination and Corruption, 2008

63. Holte, Martin Bech Essays on Incentives and Leadership, 2008

64. Damsgaard, Erika Färnstrand Essays on Technological Choice andSpillovers, 2008

65. Fredriksson, Anders Bureaucracy, Informality and Taxation: Es-says in Development Economics and Public Finance, 2009

66. Folke, Olle Parties, Power and Patronage: Papers in Political Eco-nomics, 2010

67. Drott, David Yanagizawa, Information, Markets and Con�ict: Es-says on Development and Political Economics, 2010

191

68. Meyersson, Erik, Religion, Politics and Development: Essays inDevelopment Economics and Political Economics, 2010

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71. Caldara, Dario, Essays on Empirical Macroeconomics, 2011

72. Mueller, Andreas, Business Cycles, Unemployment and Job Search:Essays in Macroeconomics and Labor Economics, 2011

73. von Below, David, Essays in Climate and Labour Economics, 2011

74. Gars, Johan, Essays on the Macroeconomics of Climate Change,2012

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Essays on the Macroeconomics of Climate Change

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